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22 lines
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nAfter the educational reform Polycarp studies only two subjects at school, Safety Studies and PE (Physical Education). During the long months of the fourth term, he received n marks in them. When teachers wrote a mark in the journal, they didn't write in what subject the mark was for, they just wrote the mark. Now it's time to show the journal to his strict parents. Polycarp knows that recently at the Parent Meeting the parents were told that he received a Safety Studies marks and b PE marks ( a \\u2009+\\u2009 b \\u2009=\\u2009 n ). Now Polycarp wants to write a subject's name in front of each mark so that: there are exactly a Safety Studies marks, there are exactly b PE marks, the total average score in both subjects is maximum. An average subject grade is the sum of all marks in it, divided by the number of them. Of course, the division is performed in real numbers without rounding up or down. Polycarp aims to maximize the x 1 \\u2009+\\u2009 x 2 , where x 1 is the average score in the first subject (Safety Studies), and x 2 is the average score in the second one (Physical Education).\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains an integer n ( 2\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 5 ), n is the number of marks in Polycarp's Journal. The second line contains two positive integers a ,\\u2009 b ( 1\\u2009\\u2264\\u2009 a ,\\u2009 b \\u2009\\u2264\\u2009 n \\u2009-\\u20091,\\u2009 a \\u2009+\\u2009 b \\u2009=\\u2009 n ). The third line contains a sequence of integers t 1 ,\\u2009 t 2 ,\\u2009...,\\u2009 t n ( 1\\u2009\\u2264\\u2009 t i \\u2009\\u2264\\u20095 ), they are Polycarp's marks.\\n\\n## Output Specification\\n\\nPrint the sequence of integers f 1 ,\\u2009 f 2 ,\\u2009...,\\u2009 f n , where f i ( 1\\u2009\\u2264\\u2009 f i \\u2009\\u2264\\u20092 ) is the number of a subject to which the i -th mark should be attributed. If there are several possible solutions, then print such that the sequence f 1 ,\\u2009 f 2 ,\\u2009...,\\u2009 f n is the smallest lexicographically. The sequence p 1 ,\\u2009 p 2 ,\\u2009...,\\u2009 p n is lexicographically less than q 1 ,\\u2009 q 2 ,\\u2009...,\\u2009 q n if there exists such j ( 1\\u2009\\u2264\\u2009 j \\u2009\\u2264\\u2009 n ) that p i \\u2009=\\u2009 q i for all 1\\u2009\\u2264\\u2009 i \\u2009<\\u2009 j , \\u0430nd p j \\u2009<\\u2009 q j .\\n\\n## Examples\\n\\n### Input #1\\n5\\n3 2\\n4 4 5 4 4\\n\\n### Output #1\\n1 1 2 1 2 \\n\\n### Input #2\\n4\\n2 2\\n3 5 4 5\\n\\n### Output #2\\n1 1 2 2 \\n\\n### Input #3\\n6\\n1 5\\n4 4 4 5 4 4\\n\\n### Output #3\\n2 2 2 1 2 2\\n\\n## Note\\n\\nIn the first sample the average score in the first subject is equal to 4, and in the second one \\u2014 to 4.5. The total average score is 8.5.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to distribute $n$ marks between two subjects, Safety Studies (Subject 1) and PE (Subject 2). We are given the total number of marks $n$, the number of marks for Subject 1 ($a$), and the number of marks for Subject 2 ($b$), where $a + b = n$. We are also given the sequence of marks $t_1, t_2, \\\\dots, t_n$. We need to assign each mark to either Subject 1 or Subject 2 such that Subject 1 gets exactly $a$ marks and Subject 2 gets exactly $b$ marks. The goal is to maximize the sum of the average scores of the two subjects.
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nThere are $n$ students and $m$ clubs in a college. The clubs are numbered from $1$ to $m$. Each student has a potential $p_i$ and is a member of the club with index $c_i$. Initially, each student is a member of exactly one club. A technical fest starts in the college, and it will run for the next $d$ days. There is a coding competition every day in the technical fest. Every day, in the morning, exactly one student of the college leaves their club. Once a student leaves their club, they will never join any club again. Every day, in the afternoon, the director of the college will select one student from each club (in case some club has no members, nobody is selected from that club) to form a team for this day's coding competition. The strength of a team is the mex of potentials of the students in the team. The director wants to know the maximum possible strength of the team for each of the coming $d$ days. Thus, every day the director chooses such team, that the team strength is maximized. The mex of the multiset $S$ is the smallest non-negative integer that is not present in $S$. For example, the mex of the $\\\\{0, 1, 1, 2, 4, 5, 9\\\\}$ is $3$, the mex of $\\\\{1, 2, 3\\\\}$ is $0$ and the mex of $\\\\varnothing$ (empty set) is $0$.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains two integers $n$ and $m$ ($1 \\\\leq m \\\\leq n \\\\leq 5000$), the number of students and the number of clubs in college. The second line contains $n$ integers $p_1, p_2, \\\\ldots, p_n$ ($0 \\\\leq p_i < 5000$), where $p_i$ is the potential of the $i$-th student. The third line contains $n$ integers $c_1, c_2, \\\\ldots, c_n$ ($1 \\\\leq c_i \\\\leq m$), which means that $i$-th student is initially a member of the club with index $c_i$. The fourth line contains an integer $d$ ($1 \\\\leq d \\\\leq n$), number of days for which the director wants to know the maximum possible strength of the team. Each of the next $d$ lines contains an integer $k_i$ ($1 \\\\leq k_i \\\\leq n$), which means that $k_i$-th student lefts their club on the $i$-th day. It is guaranteed, that the $k_i$-th student has not left their club earlier.\\n\\n## Output Specification\\n\\nFor each of the $d$ days, print the maximum possible strength of the team on that day.\\n\\n## Examples\\n\\n### Input #1\\n\\n5 3\\n0 1 2 2 0\\n1 2 2 3 2\\n5\\n3\\n2\\n4\\n5\\n1\\n\\n\\n### Output #1\\n\\n3\\n1\\n1\\n1\\n0\\n\\n\\n### Input #2\\n\\n5 3\\n0 1 2 2 1\\n1 3 2 3 2\\n5\\n4\\n2\\n3\\n5\\n1\\n\\n\\n### Output #2\\n\\n3\\n2\\n2\\n1\\n0\\n\\n\\n### Input #3\\n\\n5 5\\n0 1 2 4 5\\n1 2 3 4 5\\n4\\n2\\n3\\n5\\n4\\n\\n\\n### Output #3\\n\\n1\\n1\\n1\\n1\\n\\n## Note\\n\\nConsider the first example: On the first day, student $3$ leaves their club. Now, the remaining students are $1$, $2$, $4$ and $5$. We can select students $1$, $2$ and $4$ to get maximum possible strength, which is $3$. Note, that we can't select students $1$, $2$ and $5$, as students $2$ and $5$ belong to the same club. Also, we can't select students $1$, $3$ and $4$, since student $3$ has left their club. On the second day, student $2$ leaves their club. Now, the remaining students are $1$, $4$ and $5$. We can select students $1$, $4$ and $5$ to get maximum possible strength, which is $1$. On the third day, the remaining students are $1$ and $5$. We can select students $1$ and $5$ to get maximum possible strength, which is $1$. On the fourth day, the remaining student is $1$. We can select stude
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nYou are given a text consisting of $n$ space-separated words. There is exactly one space character between any pair of adjacent words. There are no spaces before the first word and no spaces after the last word. The length of text is the number of letters and spaces in it. $w_i$ is the $i$-th word of text. All words consist only of lowercase Latin letters. Let's denote a segment of words $w[i..j]$ as a sequence of words $w_i, w_{i + 1}, \\\\dots, w_j$. Two segments of words $w[i_1 .. j_1]$ and $w[i_2 .. j_2]$ are considered equal if $j_1 - i_1 = j_2 - i_2$, $j_1 \\\\ge i_1$, $j_2 \\\\ge i_2$, and for every $t \\\\in [0, j_1 - i_1]$ $w_{i_1 + t} = w_{i_2 + t}$. For example, for the text \\\" to be or not to be \\\" the segments $w[1..2]$ and $w[5..6]$ are equal, they correspond to the words \\\" to be \\\". An abbreviation is a replacement of some segments of words with their first uppercase letters. In order to perform an abbreviation, you have to choose at least two non-intersecting equal segments of words, and replace each chosen segment with the string consisting of first letters of the words in the segment (written in uppercase). For example, for the text \\\" a ab a a b ab a a b c \\\" you can replace segments of words $w[2..4]$ and $w[6..8]$ with an abbreviation \\\" AAA \\\" and obtain the text \\\" a AAA b AAA b c \\\", or you can replace segments of words $w[2..5]$ and $w[6..9]$ with an abbreviation \\\" AAAB \\\" and obtain the text \\\" a AAAB AAAB c \\\". What is the minimum length of the text after at most one abbreviation?\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line of the input contains one integer $n$ ($1 \\\\le n \\\\le 300$) \\u2014 the number of words in the text. The next line contains $n$ space-separated words of the text $w_1, w_2, \\\\dots, w_n$. Each word consists only of lowercase Latin letters. It is guaranteed that the length of text does not exceed $10^5$.\\n\\n## Output Specification\\n\\nPrint one integer \\u2014 the minimum length of the text after at most one abbreviation.\\n\\n## Examples\\n\\n### Input #1\\n6\\nto be or not to be\\n\\n### Output #1\\n12\\n\\n### Input #2\\n10\\na ab a a b ab a a b c\\n\\n### Output #2\\n13\\n\\n### Input #3\\n6\\naa bb aa aa bb bb\\n\\n### Output #3\\n11\\n\\n## Note\\n\\nIn the first example you can obtain the text \\\" TB or not TB \\\". In the second example you can obtain the text \\\" a AAAB AAAB c \\\". In the third example you can obtain the text \\\" AB aa AB bb \\\".\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the minimum length of a text after applying at most one abbreviation.\\nAn abbreviation involves choosing at least two non-intersecting equal segments of words and replacing each of them with a string formed by the first uppercase letters of the words in the segment.\\n\\nLet the original text consist of $n$ words $w_1, w_2, \\\\dots, w_n$.\\nThe length of the text is the sum of lengths of all words plus the number of spaces.\\nSince there is exactly one space between adjacent words, if there are $k$ words in a text, the length is $\\\\sum |w_i| + (k-1)$.\\n\\nAn abbreviation replaces a set of equal segments.\\nLet a segment be defined by the indices $[i, j]$ of words, meaning the sequence $w_i, w_{i+1}, \\\\dots, w_j$.\\nThe length of this segment in terms of charac
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nThere is a binary string $a$ of length $n$. In one operation, you can select any prefix of $a$ with an equal number of $0$ and $1$ symbols. Then all symbols in the prefix are inverted: each $0$ becomes $1$ and each $1$ becomes $0$. For example, suppose $a=0111010000$. In the first operation, we can select the prefix of length $8$ since it has four $0$'s and four $1$'s: $[01110100]00\\\\to [10001011]00$. In the second operation, we can select the prefix of length $2$ since it has one $0$ and one $1$: $[10]00101100\\\\to [01]00101100$. It is illegal to select the prefix of length $4$ for the third operation, because it has three $0$'s and one $1$. Can you transform the string $a$ into the string $b$ using some finite number of operations (possibly, none)?\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains a single integer $t$ ($1\\\\le t\\\\le 10^4$) \\u2014 the number of test cases. The first line of each test case contains a single integer $n$ ($1\\\\le n\\\\le 3\\\\cdot 10^5$) \\u2014 the length of the strings $a$ and $b$. The following two lines contain strings $a$ and $b$ of length $n$, consisting of symbols $0$ and $1$. The sum of $n$ across all test cases does not exceed $3\\\\cdot 10^5$.\\n\\n## Output Specification\\n\\nFor each test case, output \\\" YES \\\" if it is possible to transform $a$ into $b$, or \\\" NO \\\" if it is impossible. You can print each letter in any case (upper or lower).\\n\\n## Examples\\n\\n### Input #1\\n\\n5\\n10\\n0111010000\\n0100101100\\n4\\n0000\\n0000\\n3\\n001\\n000\\n12\\n010101010101\\n100110011010\\n6\\n000111\\n110100\\n\\n\\n### Output #1\\n\\nYES\\nYES\\nNO\\nYES\\nNO\\n\\n## Note\\n\\nThe first test case is shown in the statement. In the second test case, we transform $a$ into $b$ by using zero operations. In the third test case, there is no legal operation, so it is impossible to transform $a$ into $b$. In the fourth test case, here is one such transformation: Select the length $2$ prefix to get $100101010101$. Select the length $12$ prefix to get $011010101010$. Select the length $8$ prefix to get $100101011010$. Select the length $4$ prefix to get $011001011010$. Select the length $6$ prefix to get $100110011010$. In the fifth test case, the only legal operation is to transform $a$ into $111000$. From there, the only legal operation is to return to the string we started with, so we cannot transform $a$ into $b$.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks whether we can transform binary string $a$ into binary string $b$ using a specific operation.\\nThe operation allows us to select a prefix of $a$ such that the number of 0s equals the number of 1s in that prefix. Then, we invert all bits in that prefix (0 becomes 1, 1 becomes 0).\\n\\nLet's analyze the operation.\\nSuppose the string is $a_1 a_2 \\\\dots a_n$.\\nWe can choose a prefix of length $2k$ if it has $k$ zeros and $k$ ones.\\nInverting the prefix means $a_i \\\\leftarrow 1 - a_i$ for $1 \\\\le i \\\\le 2k$.\\n\\nWe want to check if $a$ can become $b$.\\nLet's consider the difference between $a$ and $b$.\\nIf $a_i = b_i$, we don't want to change $a_i$ (or change it an even number of times).\\nIf $a_i \\\\neq b_i$, we want to change $a_i$ (or change it an odd number of times).\\nLet $c_i = a_i \\\\oplus b_i$. We want to make all $c_i$ equal
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nOne day Qwerty the Ranger witnessed two transport ships collide with each other. As a result, all contents of their cargo holds scattered around the space. And now Qwerty wants to pick as many lost items as possible to sell them later. The thing is, both ships had lots of new gravitational grippers, transported to sale. A gripper is a device that can be installed on a spaceship and than draw items in space to itself (\\\"grip\\\") and transport them to the ship's cargo hold. Overall the crashed ships lost n gravitational grippers: the i -th gripper is located at a point with coordinates ( x i ,\\u2009 y i ) . Each gripper has two features \\u2014 p i (the power) and r i (the action radius) and can grip any items with mass of no more than p i at distance no more than r i . A gripper itself is an item, too and it has its mass of m i . Qwerty's ship is located at point ( x ,\\u2009 y ) and has an old magnetic gripper installed, its characteristics are p and r . There are no other grippers in the ship's cargo holds. Find the largest number of grippers Qwerty can get hold of. As he picks the items, he can arbitrarily install any gripper in the cargo hold of the ship, including the gripper he has just picked. At any moment of time the ship can have only one active gripper installed. We consider all items and the Qwerty's ship immobile when the ranger picks the items, except for when the gripper moves an item \\u2014 then the item moves to the cargo holds and the ship still remains immobile. We can assume that the ship's cargo holds have enough room for all grippers. Qwerty can use any gripper he finds or the initial gripper an arbitrary number of times.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 4 seconds\\nMemory Limit: 512 megabytes\\n\\n## Input Specification\\n\\nThe first line contains five integers x , y , p , r and n ( \\u2009-\\u200910 9 \\u2009\\u2264\\u2009 x ,\\u2009 y \\u2009\\u2264\\u200910 9 , 1\\u2009\\u2264\\u2009 p ,\\u2009 r \\u2009\\u2264\\u200910 9 , 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u2009250000 ) \\u2014 the ship's initial position, the initial gripper's features and the number of grippers that got into the space during the collision. Next n lines contain the grippers' descriptions: the i -th line contains five integers x i , y i , m i , p i , r i ( \\u2009-\\u200910 9 \\u2009\\u2264\\u2009 x i ,\\u2009 y i \\u2009\\u2264\\u200910 9 , 1\\u2009\\u2264\\u2009 m i ,\\u2009 p i ,\\u2009 r i \\u2009\\u2264\\u200910 9 ) \\u2014 the i -th gripper's coordinates and features. It is guaranteed that all grippers are located at different points. No gripper is located at the same point with Qwerty's ship.\\n\\n## Output Specification\\n\\nPrint a single number \\u2014 the maximum number of grippers Qwerty can draw to his ship. You do not need to count the initial old magnet gripper.\\n\\n## Examples\\n\\n### Input #1\\n0 0 5 10 5\\n5 4 7 11 5\\n-7 1 4 7 8\\n0 2 13 5 6\\n2 -3 9 3 4\\n13 5 1 9 9\\n\\n### Output #1\\n3\\n\\n## Note\\n\\nIn the first sample you should get the second gripper, then use the second gripper to get the first one, then use the first gripper to get the fourth one. You cannot get neither the third gripper as it is too heavy, nor the fifth one as it is too far away.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the maximum number of grippers Qwerty can collect.\\nWe start with a ship at $(x, y)$ equipped w
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nThere are $n$ robbers at coordinates $(a_1, b_1)$, $(a_2, b_2)$, ..., $(a_n, b_n)$ and $m$ searchlight at coordinates $(c_1, d_1)$, $(c_2, d_2)$, ..., $(c_m, d_m)$. In one move you can move each robber to the right (increase $a_i$ of each robber by one) or move each robber up (increase $b_i$ of each robber by one). Note that you should either increase all $a_i$ or all $b_i$, you can't increase $a_i$ for some points and $b_i$ for some other points. Searchlight $j$ can see a robber $i$ if $a_i \\\\leq c_j$ and $b_i \\\\leq d_j$. A configuration of robbers is safe if no searchlight can see a robber (i.e. if there is no pair $i,j$ such that searchlight $j$ can see a robber $i$). What is the minimum number of moves you need to perform to reach a safe configuration?\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line of input contains two integers $n$ and $m$ ($1 \\\\leq n, m \\\\leq 2000$): the number of robbers and the number of searchlight. Each of the next $n$ lines contains two integers $a_i$, $b_i$ ($0 \\\\leq a_i, b_i \\\\leq 10^6$), coordinates of robbers. Each of the next $m$ lines contains two integers $c_i$, $d_i$ ($0 \\\\leq c_i, d_i \\\\leq 10^6$), coordinates of searchlights.\\n\\n## Output Specification\\n\\nPrint one integer: the minimum number of moves you need to perform to reach a safe configuration.\\n\\n## Examples\\n\\n### Input #1\\n\\n1 1\\n0 0\\n2 3\\n\\n\\n### Output #1\\n\\n3\\n\\n\\n### Input #2\\n\\n2 3\\n1 6\\n6 1\\n10 1\\n1 10\\n7 7\\n\\n\\n### Output #2\\n\\n4\\n\\n\\n### Input #3\\n\\n1 2\\n0 0\\n0 0\\n0 0\\n\\n\\n### Output #3\\n\\n1\\n\\n\\n### Input #4\\n\\n7 3\\n0 8\\n3 8\\n2 7\\n0 10\\n5 5\\n7 0\\n3 5\\n6 6\\n3 11\\n11 5\\n\\n\\n### Output #4\\n\\n6\\n\\n## Note\\n\\nIn the first test, you can move each robber to the right three times. After that there will be one robber in the coordinates $(3, 0)$. The configuration of the robbers is safe, because the only searchlight can't see the robber, because it is in the coordinates $(2, 3)$ and $3 > 2$. In the second test, you can move each robber to the right two times and two times up. After that robbers will be in the coordinates $(3, 8)$, $(8, 3)$. It's easy the see that the configuration of the robbers is safe. It can be proved that you can't reach a safe configuration using no more than $3$ moves.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks for the minimum number of moves to make all robbers \\\"safe\\\" from all searchlights.\\nThere are $n$ robbers and $m$ searchlights.\\nEach robber $i$ is at $(a_i, b_i)$.\\nEach searchlight $j$ is at $(c_j, d_j)$.\\nA move consists of increasing all $a_i$ by 1 (move right) OR increasing all $b_i$ by 1 (move up).\\nA searchlight $j$ sees robber $i$ if $a_i \\\\leq c_j$ and $b_i \\\\leq d_j$.\\nWe want to reach a state where for all $i, j$, it is NOT the case that ($a_i \\\\leq c_j$ and $b_i \\\\leq d_j$).\\nLet $x$ be the number of moves to the right, and $y$ be the number of moves up.\\nThe total number of moves is $x + y$.\\nAfter $x$ moves right and $y$ moves up, the new coordinates of robber $i$ are $(a_i + x, b_i + y)$.\\nThe condition that robber $i$ is NOT seen by searchlight $j$ is:\\n$\\\\neg (a_i + x \\\\leq c_j \\\\text{ and } b_i + y \\\\leq d_j)$\\n$\\\\iff a_i + x > c_j \\\\text{ or } b_i + y > d_j$\\n$\\\\iff x > c_j - a_i \\\\text{ or }
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nWhile Vasya finished eating his piece of pizza, the lesson has already started. For being late for the lesson, the teacher suggested Vasya to solve one interesting problem. Vasya has an array a and integer x . He should find the number of different ordered pairs of indexes ( i ,\\u2009 j ) such that a i \\u2009\\u2264\\u2009 a j and there are exactly k integers y such that a i \\u2009\\u2264\\u2009 y \\u2009\\u2264\\u2009 a j and y is divisible by x . In this problem it is meant that pair ( i ,\\u2009 j ) is equal to ( j ,\\u2009 i ) only if i is equal to j . For example pair (1,\\u20092) is not the same as (2,\\u20091) .\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains 3 integers n ,\\u2009 x ,\\u2009 k ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 5 ,\\u20091\\u2009\\u2264\\u2009 x \\u2009\\u2264\\u200910 9 ,\\u20090\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u200910 9 ), where n is the size of the array a and x and k are numbers from the statement. The second line contains n integers a i ( 1\\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u200910 9 )\\u00a0\\u2014 the elements of the array a .\\n\\n## Output Specification\\n\\nPrint one integer\\u00a0\\u2014 the answer to the problem.\\n\\n## Examples\\n\\n### Input #1\\n4 2 1\\n1 3 5 7\\n\\n### Output #1\\n3\\n\\n### Input #2\\n4 2 0\\n5 3 1 7\\n\\n### Output #2\\n4\\n\\n### Input #3\\n5 3 1\\n3 3 3 3 3\\n\\n### Output #3\\n25\\n\\n## Note\\n\\nIn first sample there are only three suitable pairs of indexes\\u00a0\\u2014 (1,\\u20092),\\u2009(2,\\u20093),\\u2009(3,\\u20094) . In second sample there are four suitable pairs of indexes (1,\\u20091),\\u2009(2,\\u20092),\\u2009(3,\\u20093),\\u2009(4,\\u20094) . In third sample every pair ( i ,\\u2009 j ) is suitable, so the answer is 5\\u2009*\\u20095\\u2009=\\u200925 .\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the number of pairs of indices $(i, j)$ such that $a_i \\\\le a_j$ and the number of integers $y$ satisfying $a_i \\\\le y \\\\le a_j$ and $y$ is divisible by $x$ is exactly $k$.\\nThe constraints are:\\n$n \\\\le 10^5$\\n$x \\\\le 10^9$\\n$k \\\\le 10^9$\\n$a_i \\\\le 10^9$\\n\\nLet $count(L, R)$ be the number of multiples of $x$ in the range $[L, R]$.\\nThe condition is $count(a_i, a_j) = k$.\\nWe are given that $a_i \\\\le a_j$.\\nThe number of multiples of $x$ in $[L, R]$ is given by $\\\\lfloor \\\\frac{R}{x} \\\\rfloor - \\\\lfloor \\\\frac{L-1}{x} \\\\rfloor$.\\nSo the condition becomes $\\\\lfloor \\\\frac{a_j}{x} \\\\rfloor - \\\\lfloor \\\\frac{a_i-1}{x} \\\\rfloor = k$.\\n\\nLet $f(v) = \\\\lfloor \\\\frac{v}{x} \\\\rfloor$. The condition is $f(a_j) - f(a_i-1) = k$.\\nNote that $f(a_i-1)$ is the number of multiples of $x$ strictly less than $a_i$.\\nLet $val(v) = \\\\lfloor \\\\frac{v}{x} \\\\rfloor$.\\nWe need to count pairs $(i, j)$ such that $a_i \\\\le a_j$ and $val(a_j) - val(a_i-1) = k$.\\n\\nLet's analyze the term $val(a_i-1)$.\\nIf $a_i$ is a multiple of $x$, say $a_i = m \\\\cdot x$, then $a_i-1 = m \\\\cdot x - 1$.\\n$val(a_i-1) = \\\\lfloor \\\\frac{m \\\\cdot x - 1}{x} \\\\rfloor = m - 1$.\\nAlso $val(a_i) = m$.\\nSo $val(a_i-1) = val(a_i) - 1$ if $a_i \\\\% x == 0$.\\nIf $a_i$ is not a multiple of $x$, say $a_i = m \\\\cdot x + r$ where $1 \\\\le r < x$.\\nThen $a_i-1 = m \\\\cdot x + r - 1$.\\n$val(a_i-1) = m$.\\nAlso $
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\n$n$ players are playing a game. There are two different maps in the game. For each player, we know his strength on each map. When two players fight on a specific map, the player with higher strength on that map always wins. No two players have the same strength on the same map. You are the game master and want to organize a tournament. There will be a total of $n-1$ battles. While there is more than one player in the tournament, choose any map and any two remaining players to fight on it. The player who loses will be eliminated from the tournament. In the end, exactly one player will remain, and he is declared the winner of the tournament. For each player determine if he can win the tournament.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains a single integer $t$ ($1 \\\\le t \\\\le 100$) \\u2014 the number of test cases. The description of test cases follows. The first line of each test case contains a single integer $n$ ($1 \\\\leq n \\\\leq 10^5$) \\u2014 the number of players. The second line of each test case contains $n$ integers $a_1, a_2, \\\\dots, a_n$ ($1 \\\\leq a_i \\\\leq 10^9$, $a_i \\\\neq a_j$ for $i \\\\neq j$), where $a_i$ is the strength of the $i$-th player on the first map. The third line of each test case contains $n$ integers $b_1, b_2, \\\\dots, b_n$ ($1 \\\\leq b_i \\\\leq 10^9$, $b_i \\\\neq b_j$ for $i \\\\neq j$), where $b_i$ is the strength of the $i$-th player on the second map. It is guaranteed that the sum of $n$ over all test cases does not exceed $10^5$.\\n\\n## Output Specification\\n\\nFor each test case print a string of length $n$. $i$-th character should be \\\" 1 \\\" if the $i$-th player can win the tournament, or \\\" 0 \\\" otherwise.\\n\\n## Examples\\n\\n### Input #1\\n\\n3\\n4\\n1 2 3 4\\n1 2 3 4\\n4\\n11 12 20 21\\n44 22 11 30\\n1\\n1000000000\\n1000000000\\n\\n\\n### Output #1\\n\\n0001\\n1111\\n1\\n\\n## Note\\n\\nIn the first test case, the $4$-th player will beat any other player on any game, so he will definitely win the tournament. In the second test case, everyone can be a winner. In the third test case, there is only one player. Clearly, he will win the tournament.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to determine, for each player, if it's possible for them to win a tournament.\\nThere are $n$ players.\\nThere are two maps.\\nFor each player $i$, we have strength $a_i$ on map 1 and $b_i$ on map 2.\\nWhen two players fight on map 1, the one with higher $a$ wins.\\nWhen two players fight on map 2, the one with higher $b$ wins.\\nThe tournament consists of $n-1$ matches. In each match, we pick two remaining players and a map. The loser is eliminated. The last remaining player wins.\\nWe need to output a binary string where the $i$-th character is '1' if player $i$ can win, and '0' otherwise.\\n\\nLet's analyze the condition for a player to win.\\nA player $i$ wins if there exists a sequence of matches such that player $i$ is never eliminated.\\nSince we can choose the map and the pair of players for each match, we have a lot of freedom.\\nHowever, the constraint is that if player $i$ fights player $j$ on map 1, $i$ wins iff $a_i > a_j$. If they fight on map 2, $i$ wins iff $b_i > b_j$.\\nEssentially, player $i$ can defeat player $j$ if $a_i > a_j$ OR $b_i > b_j$.\\nWait, is that correct? The pr
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nYou have a long stick, consisting of $m$ segments enumerated from $1$ to $m$. Each segment is $1$ centimeter long. Sadly, some segments are broken and need to be repaired. You have an infinitely long repair tape. You want to cut some pieces from the tape and use them to cover all of the broken segments. To be precise, a piece of tape of integer length $t$ placed at some position $s$ will cover segments $s, s+1, \\\\ldots, s+t-1$. You are allowed to cover non-broken segments; it is also possible that some pieces of tape will overlap. Time is money, so you want to cut at most $k$ continuous pieces of tape to cover all the broken segments. What is the minimum total length of these pieces?\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains three integers $n$, $m$ and $k$ ($1 \\\\le n \\\\le 10^5$, $n \\\\le m \\\\le 10^9$, $1 \\\\le k \\\\le n$)\\u00a0\\u2014 the number of broken segments, the length of the stick and the maximum number of pieces you can use. The second line contains $n$ integers $b_1, b_2, \\\\ldots, b_n$ ($1 \\\\le b_i \\\\le m$)\\u00a0\\u2014 the positions of the broken segments. These integers are given in increasing order, that is, $b_1 < b_2 < \\\\ldots < b_n$.\\n\\n## Output Specification\\n\\nPrint the minimum total length of the pieces.\\n\\n## Examples\\n\\n### Input #1\\n4 100 2\\n20 30 75 80\\n\\n### Output #1\\n17\\n\\n### Input #2\\n5 100 3\\n1 2 4 60 87\\n\\n### Output #2\\n6\\n\\n## Note\\n\\nIn the first example, you can use a piece of length $11$ to cover the broken segments $20$ and $30$, and another piece of length $6$ to cover $75$ and $80$, for a total length of $17$. In the second example, you can use a piece of length $4$ to cover broken segments $1$, $2$ and $4$, and two pieces of length $1$ to cover broken segments $60$ and $87$.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the minimum total length of tape needed to cover all $n$ broken segments, using at most $k$ pieces of tape. The broken segments are given as positions $b_1, b_2, \\\\ldots, b_n$ on a stick of length $m$. The positions are sorted.\\n\\nLet's denote the broken segments as points on a number line. We want to cover these points with at most $k$ intervals.\\nIf we use 1 piece of tape to cover all broken segments, the tape must cover the range from $b_1$ to $b_n$. The length of this tape would be $b_n - b_1 + 1$.\\nIf we are allowed to use more pieces, we can \\\"break\\\" the coverage into separate intervals.\\nSuppose we decide to use exactly $k$ pieces. To minimize the total length, we should cover contiguous groups of broken segments with single pieces.\\nConsider the gaps between adjacent broken segments. The gap between $b_i$ and $b_{i+1}$ is $b_{i+1} - b_i - 1$.\\nIf we cover $b_i$ and $b_{i+1}$ with the same piece of tape, we must also cover all the non-broken segments between them. The cost contributed by this gap is effectively the length of the gap plus 1 (since we are connecting $b_i$ to $b_{i+1}$). Or, thinking about it differently:\\nThe total length of tape needed to cover a single contiguous block of broken segments from $b_i$ to $b_j$ is $b_j - b_i + 1$.\\nIf we have a set of broken segments and we want to cover them with 1 piece, the length is $b_n - b_1 + 1$.\\nIf we split this set into two sets, say $\\\\{b_1, \\\\ldots, b_i\\
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nSquirrel Liss lived in a forest peacefully, but unexpected trouble happens. Stones fall from a mountain. Initially Squirrel Liss occupies an interval [0,\\u20091] . Next, n stones will fall and Liss will escape from the stones. The stones are numbered from 1 to n in order. The stones always fall to the center of Liss's interval. When Liss occupies the interval [ k \\u2009-\\u2009 d ,\\u2009 k \\u2009+\\u2009 d ] and a stone falls to k , she will escape to the left or to the right. If she escapes to the left, her new interval will be [ k \\u2009-\\u2009 d ,\\u2009 k ] . If she escapes to the right, her new interval will be [ k ,\\u2009 k \\u2009+\\u2009 d ] . You are given a string s of length n . If the i -th character of s is \\\" l \\\" or \\\" r \\\", when the i -th stone falls Liss will escape to the left or to the right, respectively. Find the sequence of stones' numbers from left to right after all the n stones falls.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe input consists of only one line. The only line contains the string s ( 1\\u2009\\u2264\\u2009| s |\\u2009\\u2264\\u200910 6 ). Each character in s will be either \\\" l \\\" or \\\" r \\\".\\n\\n## Output Specification\\n\\nOutput n lines \\u2014 on the i -th line you should print the i -th stone's number from the left.\\n\\n## Examples\\n\\n### Input #1\\nllrlr\\n\\n### Output #1\\n3\\n5\\n4\\n2\\n1\\n\\n### Input #2\\nrrlll\\n\\n### Output #2\\n1\\n2\\n5\\n4\\n3\\n\\n### Input #3\\nlrlrr\\n\\n### Output #3\\n2\\n4\\n5\\n3\\n1\\n\\n## Note\\n\\nIn the first example, the positions of stones 1, 2, 3, 4, 5 will be , respectively. So you should print the sequence: 3, 5, 4, 2, 1.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to simulate a process where stones fall into an interval, and the interval is split. We need to determine the order of stones from left to right after all $n$ stones have fallen.\\n\\nLet's trace the process.\\nInitially, the interval is $[0, 1]$.\\nThere are $n$ stones, numbered $1$ to $n$.\\nA string $s$ of length $n$ dictates the movement. $s[i]$ corresponds to the $i$-th stone falling.\\nWait, the problem says \\\"stones are numbered from 1 to n in order\\\". And \\\"The stones always fall to the center of Liss's interval\\\".\\nLet's look at the example to understand the positions.\\nExample 1: `llrlr` ($n=5$).\\nInitially interval $[0, 1]$. Center is $0.5$.\\nStone 1 falls at $0.5$.\\n$s[0]$ is 'l'. Liss escapes to the left. New interval $[0, 0.5]$.\\nStone 2 falls at center of $[0, 0.5]$, which is $0.25$.\\n$s[1]$ is 'l'. Liss escapes to the left. New interval $[0, 0.25]$.\\nStone 3 falls at center of $[0, 0.25]$, which is $0.125$.\\n$s[2]$ is 'r'. Liss escapes to the right. New interval $[0.125, 0.25]$.\\nStone 4 falls at center of $[0.125, 0.25]$, which is $0.1875$.\\n$s[3]$ is 'l'. Liss escapes to the left. New interval $[0.125, 0.1875]$.\\nStone 5 falls at center of $[0.125, 0.1875]$, which is $0.15625$.\\n$s[4]$ is 'r'. Liss escapes to the right. New interval $[0.15625, 0.1875]$.\\n\\nThe positions of stones are:\\n1: 0.5\\n2: 0.25\\n3: 0.125\\n4: 0.1875\\n5: 0.15625\\n\\nSorting these positions:\\n0.125 (Stone 3)\\n0.15625 (Stone 5)\\n0.1875 (Stone 4)\\n0.25 (Stone 2)\\n0.5 (Stone 1)\\nOrder: 3, 5, 4, 2, 1.\\nThis matches Example 1 output.\\n\\nLet's look at the structure.\\n
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nSanta has to send presents to the kids. He has a large stack of $n$ presents, numbered from $1$ to $n$; the topmost present has number $a_1$, the next present is $a_2$, and so on; the bottom present has number $a_n$. All numbers are distinct. Santa has a list of $m$ distinct presents he has to send: $b_1$, $b_2$, ..., $b_m$. He will send them in the order they appear in the list . To send a present, Santa has to find it in the stack by removing all presents above it, taking this present and returning all removed presents on top of the stack. So, if there are $k$ presents above the present Santa wants to send, it takes him $2k + 1$ seconds to do it. Fortunately, Santa can speed the whole process up \\u2014 when he returns the presents to the stack, he may reorder them as he wishes (only those which were above the present he wanted to take; the presents below cannot be affected in any way). What is the minimum time required to send all of the presents, provided that Santa knows the whole list of presents he has to send and reorders the presents optimally? Santa cannot change the order of presents or interact with the stack of presents in any other way. Your program has to answer $t$ different test cases.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains one integer $t$ ($1 \\\\le t \\\\le 100$) \\u2014 the number of test cases. Then the test cases follow, each represented by three lines. The first line contains two integers $n$ and $m$ ($1 \\\\le m \\\\le n \\\\le 10^5$) \\u2014 the number of presents in the stack and the number of presents Santa wants to send, respectively. The second line contains $n$ integers $a_1$, $a_2$, ..., $a_n$ ($1 \\\\le a_i \\\\le n$, all $a_i$ are unique) \\u2014 the order of presents in the stack. The third line contains $m$ integers $b_1$, $b_2$, ..., $b_m$ ($1 \\\\le b_i \\\\le n$, all $b_i$ are unique) \\u2014 the ordered list of presents Santa has to send. The sum of $n$ over all test cases does not exceed $10^5$.\\n\\n## Output Specification\\n\\nFor each test case print one integer \\u2014 the minimum number of seconds which Santa has to spend sending presents, if he reorders the presents optimally each time he returns them into the stack.\\n\\n## Examples\\n\\n### Input #1\\n\\n2\\n3 3\\n3 1 2\\n3 2 1\\n7 2\\n2 1 7 3 4 5 6\\n3 1\\n\\n\\n### Output #1\\n\\n5\\n8\\n\\n## Note\\n\\nNone\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to calculate the minimum time required for Santa to send $m$ presents from a stack of $n$ presents. The stack is given as a list $a_1, a_2, \\\\dots, a_n$, where $a_1$ is the top. Santa needs to send presents $b_1, b_2, \\\\dots, b_m$ in that specific order.\\n\\nTo send a present $x$, Santa must remove all presents above it. If there are $k$ presents above $x$, it takes $2k + 1$ seconds. After taking $x$, Santa puts the $k$ presents back on top of the stack. Crucially, he can reorder these $k$ presents however he wants. He wants to minimize the total time.\\n\\nLet's analyze the cost. If the present to be sent is at position $pos$ (1-indexed from the top), there are $pos-1$ presents above it. The cost is $2(pos-1) + 1 = 2 \\\\cdot pos - 1$.\\nSince Santa can reorder the presents he puts back, he can choose to place the presents that he will need to send *later* on top of the stack, in the order
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nA and B are preparing themselves for programming contests. An important part of preparing for a competition is sharing programming knowledge from the experienced members to those who are just beginning to deal with the contests. Therefore, during the next team training A decided to make teams so that newbies are solving problems together with experienced participants. A believes that the optimal team of three people should consist of one experienced participant and two newbies. Thus, each experienced participant can share the experience with a large number of people. However, B believes that the optimal team should have two experienced members plus one newbie. Thus, each newbie can gain more knowledge and experience. As a result, A and B have decided that all the teams during the training session should belong to one of the two types described above. Furthermore, they agree that the total number of teams should be as much as possible. There are n experienced members and m newbies on the training session. Can you calculate what maximum number of teams can be formed?\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains two integers n and m ( 0\\u2009\\u2264\\u2009 n ,\\u2009 m \\u2009\\u2264\\u20095\\u00b710 5 ) \\u2014 the number of experienced participants and newbies that are present at the training session.\\n\\n## Output Specification\\n\\nPrint the maximum number of teams that can be formed.\\n\\n## Examples\\n\\n### Input #1\\n2 6\\n\\n### Output #1\\n2\\n\\n### Input #2\\n4 5\\n\\n### Output #2\\n3\\n\\n## Note\\n\\nLet's represent the experienced players as XP and newbies as NB. In the first test the teams look as follows: (XP, NB, NB), (XP, NB, NB). In the second test sample the teams look as follows: (XP, NB, NB), (XP, NB, NB), (XP, XP, NB).\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the maximum number of teams that can be formed given $n$ experienced members and $m$ newbies.\\nThere are two types of teams:\\n1. 1 experienced member + 2 newbies (1X, 2N)\\n2. 2 experienced members + 1 newbie (2X, 1N)\\n\\nWe want to maximize the total number of teams.\\nLet $x$ be the number of teams of type 1.\\nLet $y$ be the number of teams of type 2.\\n\\nThe constraints on the number of people are:\\n$x \\\\cdot 1 + y \\\\cdot 2 \\\\le n$ (experienced members used)\\n$x \\\\cdot 2 + y \\\\cdot 1 \\\\le m$ (newbies used)\\n\\nWe want to maximize $x + y$.\\nThe variables $x, y$ must be non-negative integers.\\n\\nInput limits: $n, m \\\\le 5 \\\\cdot 10^5$.\\nTime limit: 1 second.\\nMemory limit: 256 MB.\\n\\nSince $n$ and $m$ are relatively small, we might consider iterating over one variable. However, $O(n)$ or $O(m)$ is acceptable ($5 \\\\cdot 10^5$ operations is well within 1 second).\\nBut actually, this is a linear programming type problem or simply a math problem. Let's analyze the constraints.\\n\\nTotal people used = $3(x+y)$.\\nSo $3(x+y) \\\\le n + m$.\\nThus $x+y \\\\le \\\\lfloor (n+m)\\/3 \\\\rfloor$.\\nAlso, obviously, the number of teams cannot exceed the number of experienced members (since each team needs at least 1) and cannot exceed the number of newbies (since each team needs at least 1).\\nSo $x+y \\\\le n$ and $x+y \\\\le m$.\\nWait, is it true that $x+y \\\\le n$?\\nType 1 needs 1 X, Type 2 needs 2 X. So yes, total X used is $x+2
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nAllen is hosting a formal dinner party. $2n$ people come to the event in $n$ pairs (couples). After a night of fun, Allen wants to line everyone up for a final picture. The $2n$ people line up, but Allen doesn't like the ordering. Allen prefers if each pair occupies adjacent positions in the line, as this makes the picture more aesthetic. Help Allen find the minimum number of swaps of adjacent positions he must perform to make it so that each couple occupies adjacent positions in the line.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains a single integer $n$ ($1 \\\\le n \\\\le 100$), the number of pairs of people. The second line contains $2n$ integers $a_1, a_2, \\\\dots, a_{2n}$. For each $i$ with $1 \\\\le i \\\\le n$, $i$ appears exactly twice. If $a_j = a_k = i$, that means that the $j$-th and $k$-th people in the line form a couple.\\n\\n## Output Specification\\n\\nOutput a single integer, representing the minimum number of adjacent swaps needed to line the people up so that each pair occupies adjacent positions.\\n\\n## Examples\\n\\n### Input #1\\n4\\n1 1 2 3 3 2 4 4\\n\\n### Output #1\\n2\\n\\n### Input #2\\n3\\n1 1 2 2 3 3\\n\\n### Output #2\\n0\\n\\n### Input #3\\n3\\n3 1 2 3 1 2\\n\\n### Output #3\\n3\\n\\n## Note\\n\\nIn the first sample case, we can transform $1 1 2 3 3 2 4 4 \\\\rightarrow 1 1 2 3 2 3 4 4 \\\\rightarrow 1 1 2 2 3 3 4 4$ in two steps. Note that the sequence $1 1 2 3 3 2 4 4 \\\\rightarrow 1 1 3 2 3 2 4 4 \\\\rightarrow 1 1 3 3 2 2 4 4$ also works in the same number of steps. The second sample case already satisfies the constraints; therefore we need $0$ swaps.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks for the minimum number of adjacent swaps required to arrange $2n$ people, consisting of $n$ couples (where each couple is identified by a number from $1$ to $n$), such that each couple occupies adjacent positions.\\n\\nLet the array be $A$ of size $2n$.\\nWe need to transform $A$ into an array $B$ such that for every $k \\\\in \\\\{0, 2, \\\\dots, 2n-2\\\\}$, $B[k]$ and $B[k+1]$ are the same value (representing a couple).\\nThe cost of transforming $A$ to $B$ is the number of adjacent swaps, which is equivalent to the number of inversions if we map elements to their target positions, or more simply, the minimum number of swaps to reach a specific permutation is the number of inversions between the initial and final states. However, here the target state is not unique. We can arrange the couples in any order (e.g., couple 1 then couple 2, or couple 2 then couple 1, etc.), and within a couple, the order doesn't matter (e.g., $1, 1$ is the same as $1, 1$). Actually, the values are identical, so order within a couple doesn't matter.\\n\\nLet's consider the constraints. $n \\\\le 100$. This means the array size is at most 200.\\nSince $n$ is small, an $O(n^2)$ or even $O(n^3)$ solution is acceptable.\\nThe problem asks for the minimum number of swaps.\\n\\nLet's try a greedy approach. We iterate through the array from left to right. At index $i$ (where $i$ is even, $0, 2, \\\\dots, 2n-2$), we want to place a couple at positions $i$ and $i+1$.\\nSuppose we are at index $i$. The element $A[i]$ is already fixed (we can't move it to the left because we've processed $0 \\\\dots i-1$). Let $x = A[i]$. We need to find the other occurr
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nThere are $n$ cities along the road, which can be represented as a straight line. The $i$-th city is situated at the distance of $a_i$ kilometers from the origin. All cities are situated in the same direction from the origin. There are $m$ trucks travelling from one city to another. Each truck can be described by $4$ integers: starting city $s_i$, finishing city $f_i$, fuel consumption $c_i$ and number of possible refuelings $r_i$. The $i$-th truck will spend $c_i$ litres of fuel per one kilometer. When a truck arrives in some city, it can be refueled (but refueling is impossible in the middle of nowhere). The $i$-th truck can be refueled at most $r_i$ times. Each refueling makes truck's gas-tank full. All trucks start with full gas-tank. All trucks will have gas-tanks of the same size $V$ litres. You should find minimum possible $V$ such that all trucks can reach their destinations without refueling more times than allowed.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nFirst line contains two integers $n$ and $m$ ($2 \\\\le n \\\\le 400$, $1 \\\\le m \\\\le 250000$) \\u2014 the number of cities and trucks. The second line contains $n$ integers $a_1, a_2, \\\\dots, a_n$ ($1 \\\\le a_i \\\\le 10^9$, $a_i < a_{i+1}$) \\u2014 positions of cities in the ascending order. Next $m$ lines contains $4$ integers each. The $i$-th line contains integers $s_i$, $f_i$, $c_i$, $r_i$ ($1 \\\\le s_i < f_i \\\\le n$, $1 \\\\le c_i \\\\le 10^9$, $0 \\\\le r_i \\\\le n$) \\u2014 the description of the $i$-th truck.\\n\\n## Output Specification\\n\\nPrint the only integer \\u2014 minimum possible size of gas-tanks $V$ such that all trucks can reach their destinations.\\n\\n## Examples\\n\\n### Input #1\\n\\n7 6\\n2 5 7 10 14 15 17\\n1 3 10 0\\n1 7 12 7\\n4 5 13 3\\n4 7 10 1\\n4 7 10 1\\n1 5 11 2\\n\\n\\n### Output #1\\n\\n55\\n\\n## Note\\n\\nLet's look at queries in details: the $1$-st truck must arrive at position $7$ from $2$ without refuelling, so it needs gas-tank of volume at least $50$. the $2$-nd truck must arrive at position $17$ from $2$ and can be refueled at any city (if it is on the path between starting point and ending point), so it needs gas-tank of volume at least $48$. the $3$-rd truck must arrive at position $14$ from $10$, there is no city between, so it needs gas-tank of volume at least $52$. the $4$-th truck must arrive at position $17$ from $10$ and can be refueled only one time: it's optimal to refuel at $5$-th city (position $14$) so it needs gas-tank of volume at least $40$. the $5$-th truck has the same description, so it also needs gas-tank of volume at least $40$. the $6$-th truck must arrive at position $14$ from $2$ and can be refueled two times: first time in city $2$ or $3$ and second time in city $4$ so it needs gas-tank of volume at least $55$.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks for the minimum gas tank size $V$ such that all $m$ trucks can complete their trips.\\nWe have $n$ cities located at distances $a_1, a_2, \\\\dots, a_n$ from the origin.\\nThere are $m$ trucks. Each truck $i$ is defined by:\\n- Start city $s_i$ (index)\\n- Finish city $f_i$ (index)\\n- Fuel consumption per km $c_i$\\n- Max refuelings $r_i$\\n\\nAll trucks start with a full tank of size $V$.\\nRefueling is possible only at cities.\\nRefueling fills the tank to $V$.\\nThe t
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nThere are $n$ computers in a row, all originally off, and Phoenix wants to turn all of them on. He will manually turn on computers one at a time. At any point, if computer $i-1$ and computer $i+1$ are both on, computer $i$ $(2 \\\\le i \\\\le n-1)$ will turn on automatically if it is not already on. Note that Phoenix cannot manually turn on a computer that already turned on automatically. If we only consider the sequence of computers that Phoenix turns on manually, how many ways can he turn on all the computers? Two sequences are distinct if either the set of computers turned on manually is distinct, or the order of computers turned on manually is distinct. Since this number may be large, please print it modulo $M$.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 3 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains two integers $n$ and $M$ ($3 \\\\le n \\\\le 400$; $10^8 \\\\le M \\\\le 10^9$)\\u00a0\\u2014 the number of computers and the modulo. It is guaranteed that $M$ is prime.\\n\\n## Output Specification\\n\\nPrint one integer\\u00a0\\u2014 the number of ways to turn on the computers modulo $M$.\\n\\n## Examples\\n\\n### Input #1\\n\\n3 100000007\\n\\n\\n### Output #1\\n\\n6\\n\\n\\n### Input #2\\n\\n4 100000007\\n\\n\\n### Output #2\\n\\n20\\n\\n\\n### Input #3\\n\\n400 234567899\\n\\n\\n### Output #3\\n\\n20914007\\n\\n## Note\\n\\nIn the first example, these are the $6$ orders in which Phoenix can turn on all computers: $[1,3]$. Turn on computer $1$, then $3$. Note that computer $2$ turns on automatically after computer $3$ is turned on manually, but we only consider the sequence of computers that are turned on manually. $[3,1]$. Turn on computer $3$, then $1$. $[1,2,3]$. Turn on computer $1$, $2$, then $3$. $[2,1,3]$ $[2,3,1]$ $[3,2,1]$\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the number of ways to turn on $n$ computers, initially all off.\\nPhoenix manually turns on computers one by one.\\nThere is an automatic rule: if computer $i-1$ and $i+1$ are both on, computer $i$ turns on automatically (if it wasn't already on).\\nPhoenix cannot manually turn on a computer that is already on.\\nWe need to count the number of sequences of manually turned on computers such that eventually all $n$ computers are on. The answer should be modulo $M$.\\n$n \\\\le 400$, $M \\\\le 10^9$ (prime).\\n\\nLet's analyze the automatic rule.\\nComputer $i$ turns on automatically if $i-1$ and $i+1$ are on.\\nThis looks like filling gaps. If we have a segment of consecutive on computers, say $[L, R]$, and we turn on $L-1$ or $R+1$, the gap might close or extend.\\nActually, the condition is local: $i-1$ and $i+1$ ON $\\\\implies i$ ON.\\nThis means that if we have a block of ON computers, say from $L$ to $R$, and we turn on $L-1$ and $R+1$ (in any order), then all computers in between $L$ and $R$ (which are already on) stay on. Wait, if $i-1$ and $i+1$ are on, $i$ turns on.\\nConsider the state of computers as a binary string of length $n$. Initially $00\\\\dots0$.\\nWe want to reach $11\\\\dots1$.\\nThe operation is: pick an index $i$ such that $state[i] = 0$, set $state[i] = 1$. Then, repeatedly apply the rule: if $state[j-1]=1$ and $state[j+1]=1$, set $state[j]=1$.\\nThe question asks for the number of valid sequences of manual operations.\\nWait, \\\"Two sequences are distinct if either the set of c
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nKolya is developing an economy simulator game. His most favourite part of the development process is in-game testing. Once he was entertained by the testing so much, that he found out his game-coin score become equal to 0 . Kolya remembers that at the beginning of the game his game-coin score was equal to n and that he have bought only some houses (for 1\\u2009234\\u2009567 game-coins each), cars (for 123\\u2009456 game-coins each) and computers (for 1\\u2009234 game-coins each). Kolya is now interested, whether he could have spent all of his initial n game-coins buying only houses, cars and computers or there is a bug in the game. Formally, is there a triple of non-negative integers a , b and c such that a \\u2009\\u00d7\\u20091\\u2009234\\u2009567\\u2009+\\u2009 b \\u2009\\u00d7\\u2009123\\u2009456\\u2009+\\u2009 c \\u2009\\u00d7\\u20091\\u2009234\\u2009=\\u2009 n ? Please help Kolya answer this question.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line of the input contains a single integer n ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 9 )\\u00a0\\u2014 Kolya's initial game-coin score.\\n\\n## Output Specification\\n\\nPrint \\\" YES \\\" (without quotes) if it's possible that Kolya spent all of his initial n coins buying only houses, cars and computers. Otherwise print \\\" NO \\\" (without quotes).\\n\\n## Examples\\n\\n### Input #1\\n1359257\\n\\n### Output #1\\nYES\\n\\n### Input #2\\n17851817\\n\\n### Output #2\\nNO\\n\\n## Note\\n\\nIn the first sample, one of the possible solutions is to buy one house, one car and one computer, spending 1\\u2009234\\u2009567\\u2009+\\u2009123\\u2009456\\u2009+\\u20091234\\u2009=\\u20091\\u2009359\\u2009257 game-coins in total.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe user wants a C++ solution for a competitive programming problem.\\nThe problem asks if a given integer $n$ can be represented as a linear combination of three constants: $A = 1234567$, $B = 123456$, and $C = 1234$, with non-negative integer coefficients $a, b, c$.\\nEquation: $a \\\\cdot A + b \\\\cdot B + c \\\\cdot C = n$.\\nConstraints: $1 \\\\le n \\\\le 10^9$.\\nTime Limit: 1 second.\\nMemory Limit: 256 MB.\\n\\nLet's analyze the constants:\\n$A = 1234567$\\n$B = 123456$\\n$C = 1234$\\n\\nNotice that $B$ is roughly $A\\/10$ and $C$ is roughly $B\\/100$.\\nSpecifically:\\n$A = 1234567$\\n$B = 123456$\\n$C = 1234$\\n\\nWe need to find non-negative integers $a, b, c$ such that $a \\\\cdot A + b \\\\cdot B + c \\\\cdot C = n$.\\n\\nSince $n$ can be up to $10^9$, and $A \\\\approx 1.2 \\\\times 10^6$, the maximum possible value for $a$ is around $10^9 \\/ 10^6 = 1000$.\\nIf we iterate over $a$, the remaining value $n' = n - a \\\\cdot A$ must be representable as $b \\\\cdot B + c \\\\cdot C$.\\nThe maximum value for $a$ is roughly $1000$.\\nFor a fixed $a$, we need to check if $n' = b \\\\cdot B + c \\\\cdot C$ has a solution for non-negative $b, c$.\\nHere $B = 123456$ and $C = 1234$.\\nThe maximum value for $b$ would be roughly $n' \\/ B$. Since $n \\\\le 10^9$, $n' \\\\le 10^9$, so $b$ can be up to $10^9 \\/ 10^5 \\\\approx 10000$.\\nIterating over both $a$ and $b$ would be roughly $1000 \\\\times 10000 = 10^7$ operations, which fits well within 1 second (usually $\\\\sim 10^8$ ops\\/sec).\\n\\nHowever, we can optimize.\\nIf we fix $a$, we need to check if $
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nIvar the Boneless is a great leader. He is trying to capture Kattegat from Lagertha. The war has begun and wave after wave Ivar's warriors are falling in battle. Ivar has $n$ warriors, he places them on a straight line in front of the main gate, in a way that the $i$-th warrior stands right after $(i-1)$-th warrior. The first warrior leads the attack. Each attacker can take up to $a_i$ arrows before he falls to the ground, where $a_i$ is the $i$-th warrior's strength. Lagertha orders her warriors to shoot $k_i$ arrows during the $i$-th minute, the arrows one by one hit the first still standing warrior. After all Ivar's warriors fall and all the currently flying arrows fly by, Thor smashes his hammer and all Ivar's warriors get their previous strengths back and stand up to fight again. In other words, if all warriors die in minute $t$, they will all be standing to fight at the end of minute $t$. The battle will last for $q$ minutes, after each minute you should tell Ivar what is the number of his standing warriors.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains two integers $n$ and $q$ ($1 \\\\le n, q \\\\leq 200\\\\,000$)\\u00a0\\u2014 the number of warriors and the number of minutes in the battle. The second line contains $n$ integers $a_1, a_2, \\\\ldots, a_n$ ($1 \\\\leq a_i \\\\leq 10^9$) that represent the warriors' strengths. The third line contains $q$ integers $k_1, k_2, \\\\ldots, k_q$ ($1 \\\\leq k_i \\\\leq 10^{14}$), the $i$-th of them represents Lagertha's order at the $i$-th minute: $k_i$ arrows will attack the warriors.\\n\\n## Output Specification\\n\\nOutput $q$ lines, the $i$-th of them is the number of standing warriors after the $i$-th minute.\\n\\n## Examples\\n\\n### Input #1\\n5 5\\n1 2 1 2 1\\n3 10 1 1 1\\n\\n### Output #1\\n3\\n5\\n4\\n4\\n3\\n\\n### Input #2\\n4 4\\n1 2 3 4\\n9 1 10 6\\n\\n### Output #2\\n1\\n4\\n4\\n1\\n\\n## Note\\n\\nIn the first example: after the 1-st minute, the 1-st and 2-nd warriors die. after the 2-nd minute all warriors die (and all arrows left over are wasted), then they will be revived thus answer is 5\\u00a0\\u2014 all warriors are alive. after the 3-rd minute, the 1-st warrior dies. after the 4-th minute, the 2-nd warrior takes a hit and his strength decreases by 1. after the 5-th minute, the 2-nd warrior dies.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to simulate a battle process involving $n$ warriors and $q$ minutes of attacks.\\nWe have $n$ warriors arranged in a line. The $i$-th warrior has strength $a_i$.\\nIn each minute $i$, $k_i$ arrows are shot.\\nArrows hit the first standing warrior. If a warrior takes enough damage (total damage $\\\\ge a_i$), they fall. The arrows continue to hit the next standing warrior.\\nIf all warriors fall, they all revive immediately with full strength, and any remaining arrows in that minute are wasted.\\nAfter each minute, we need to output the number of standing warriors.\\n\\nConstraints:\\n$n, q \\\\le 200,000$.\\n$a_i \\\\le 10^9$.\\n$k_i \\\\le 10^{14}$.\\nTime limit: 2 seconds.\\n\\nLet's denote the current damage dealt to the first standing warrior as `current_damage`.\\nHowever, since warriors revive if all die, we need to track the state of the warriors.\\nActually, the state is simpler:\\nWe have a prefix of warriors that are dead. Let's
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nLevko loves permutations very much. A permutation of length n is a sequence of distinct positive integers, each is at most n . Let\\u2019s assume that value gcd ( a ,\\u2009 b ) shows the greatest common divisor of numbers a and b . Levko assumes that element p i of permutation p 1 ,\\u2009 p 2 ,\\u2009... ,\\u2009 p n is good if gcd ( i ,\\u2009 p i )\\u2009>\\u20091 . Levko considers a permutation beautiful , if it has exactly k good elements. Unfortunately, he doesn\\u2019t know any beautiful permutation. Your task is to help him to find at least one of them.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe single line contains two integers n and k ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 5 , 0\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u2009 n ).\\n\\n## Output Specification\\n\\nIn a single line print either any beautiful permutation or -1, if such permutation doesn\\u2019t exist. If there are multiple suitable permutations, you are allowed to print any of them.\\n\\n## Examples\\n\\n### Input #1\\n4 2\\n\\n### Output #1\\n2 4 3 1\\n\\n### Input #2\\n1 1\\n\\n### Output #2\\n-1\\n\\n## Note\\n\\nIn the first sample elements 4 and 3 are good because gcd (2,\\u20094)\\u2009=\\u20092\\u2009>\\u20091 and gcd (3,\\u20093)\\u2009=\\u20093\\u2009>\\u20091 . Elements 2 and 1 are not good because gcd (1,\\u20092)\\u2009=\\u20091 and gcd (4,\\u20091)\\u2009=\\u20091 . As there are exactly 2 good elements, the permutation is beautiful. The second sample has no beautiful permutations.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to construct a permutation $p$ of length $n$ such that there are exactly $k$ indices $i$ where $\\\\gcd(i, p_i) > 1$. If no such permutation exists, we should output -1.\\n\\nConstraints:\\n$1 \\\\le n \\\\le 10^5$\\n$0 \\\\le k \\\\le n$\\n\\nLet's analyze the condition $\\\\gcd(i, p_i) > 1$.\\nIf $i=1$, then $\\\\gcd(1, p_1) = 1$ always, since $p_1 \\\\ge 1$. So index 1 can never be a \\\"good\\\" element.\\nThis implies that if $k = n$, it's impossible, because at least index 1 is not good.\\nWait, let's check the example.\\nInput #2: $n=1, k=1$. Output: -1.\\nIndeed, for $n=1$, the only permutation is $[1]$. $\\\\gcd(1, 1) = 1$, which is not $> 1$. So 0 good elements. Since we need $k=1$, it's impossible.\\nSo, if $k=n$, output -1.\\n\\nFor any other case $k < n$, we need to construct such a permutation.\\nWe need $k$ indices $i$ where $\\\\gcd(i, p_i) > 1$.\\nThe simplest way to satisfy $\\\\gcd(i, p_i) > 1$ is to set $p_i = i$. Then $\\\\gcd(i, i) = i$. If $i > 1$, then $\\\\gcd(i, i) = i > 1$.\\nSo, if we set $p_i = i$ for $k$ indices $i > 1$, we get $k$ good elements.\\nHowever, we need to form a valid permutation.\\nIf we set $p_i = i$ for a set of indices, the remaining indices must be permuted among themselves such that none of them satisfy the condition (or satisfy it, depending on how we count).\\nActually, the problem asks for *exactly* $k$ good elements.\\nLet's try to make the first $k$ indices (excluding 1) have $p_i = i$, or something similar.\\nBut wait, index 1 is special. $\\\\gcd(1, p_1) = 1$. So index 1 is never good.\\nIf we want $k$ good elements, we should pick $k$ indices from $\\\\{2, 3, \\\\dots, n\\\\}$ to be good.\\nThe easiest way to make index $i$ good is to set $p_i = i$. Then $\\\\gcd(i, i) = i > 1$ (since $i \\\\g
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nSuppose that you are in a campus and have to go for classes day by day. As you may see, when you hurry to a classroom, you surprisingly find that many seats there are already occupied. Today you and your friends went for class, and found out that some of the seats were occupied. The classroom contains $n$ rows of seats and there are $m$ seats in each row. Then the classroom can be represented as an $n \\\\times m$ matrix. The character ' . ' represents an empty seat, while ' * ' means that the seat is occupied. You need to find $k$ consecutive empty seats in the same row or column and arrange those seats for you and your friends. Your task is to find the number of ways to arrange the seats. Two ways are considered different if sets of places that students occupy differs.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains three positive integers $n,m,k$ ($1 \\\\leq n, m, k \\\\leq 2\\\\,000$), where $n,m$ represent the sizes of the classroom and $k$ is the number of consecutive seats you need to find. Each of the next $n$ lines contains $m$ characters ' . ' or ' * '. They form a matrix representing the classroom, ' . ' denotes an empty seat, and ' * ' denotes an occupied seat.\\n\\n## Output Specification\\n\\nA single number, denoting the number of ways to find $k$ empty seats in the same row or column.\\n\\n## Examples\\n\\n### Input #1\\n2 3 2\\n**.\\n...\\n\\n### Output #1\\n3\\n\\n### Input #2\\n1 2 2\\n..\\n\\n### Output #2\\n1\\n\\n### Input #3\\n3 3 4\\n.*.\\n*.*\\n.*.\\n\\n### Output #3\\n0\\n\\n## Note\\n\\nIn the first sample, there are three ways to arrange those seats. You can take the following seats for your arrangement. $(1,3)$, $(2,3)$ $(2,2)$, $(2,3)$ $(2,1)$, $(2,2)$\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to count the number of ways to choose $k$ consecutive empty seats in a grid of size $n \\\\times m$. The seats can be chosen either in the same row or in the same column. The grid is given as a matrix of characters where '.' represents an empty seat and '*' represents an occupied seat.\\n\\nInput constraints:\\n$1 \\\\leq n, m, k \\\\leq 2000$.\\nTime limit: 1 second.\\nMemory limit: 256 MB.\\n\\nLet's denote the grid as $A$, where $A[i][j]$ is the character at row $i$ and column $j$ (0-indexed or 1-indexed).\\nWe need to find the number of segments of length $k$ consisting entirely of '.' characters.\\nA segment is defined by a starting position and a direction (horizontal or vertical).\\nSince the problem asks for the number of *sets* of places, and a set of places corresponds to a specific segment, we just need to count valid segments.\\n\\nWait, let's double check if \\\"sets of places\\\" implies that overlapping sets are distinct.\\n\\\"Two ways are considered different if sets of places that students occupy differs.\\\"\\nYes, a set of places is determined by the coordinates. Since a segment of length $k$ is a set of $k$ coordinates, two segments are different if they contain at least one different coordinate.\\nFor example, in a row `...`, if $k=2$, the segments are indices $\\\\{0, 1\\\\}$ and $\\\\{1, 2\\\\}$. These are distinct sets.\\nSo we just need to count the number of valid horizontal segments of length $k$ and the number of valid vertical segments of length $k$.\\n\\nLet's analyze the complexity.\\n$n, m \\\\l
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nHarry, Ron and Hermione have figured out that Helga Hufflepuff's cup is a horcrux. Through her encounter with Bellatrix Lestrange, Hermione came to know that the cup is present in Bellatrix's family vault in Gringott's Wizarding Bank. The Wizarding bank is in the form of a tree with total n vaults where each vault has some type, denoted by a number between 1 to m . A tree is an undirected connected graph with no cycles. The vaults with the highest security are of type k , and all vaults of type k have the highest security. There can be at most x vaults of highest security . Also, if a vault is of the highest security, its adjacent vaults are guaranteed to not be of the highest security and their type is guaranteed to be less than k . Harry wants to consider every possibility so that he can easily find the best path to reach Bellatrix's vault. So, you have to tell him, given the tree structure of Gringotts, the number of possible ways of giving each vault a type such that the above conditions hold.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line of input contains two space separated integers, n and m \\u00a0\\u2014 the number of vaults and the number of different vault types possible. ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 5 ,\\u20091\\u2009\\u2264\\u2009 m \\u2009\\u2264\\u200910 9 ). Each of the next n \\u2009-\\u20091 lines contain two space separated integers u i and v i ( 1\\u2009\\u2264\\u2009 u i ,\\u2009 v i \\u2009\\u2264\\u2009 n ) representing the i -th edge, which shows there is a path between the two vaults u i and v i . It is guaranteed that the given graph is a tree. The last line of input contains two integers k and x ( 1\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u2009 m ,\\u20091\\u2009\\u2264\\u2009 x \\u2009\\u2264\\u200910 ), the type of the highest security vault and the maximum possible number of vaults of highest security.\\n\\n## Output Specification\\n\\nOutput a single integer, the number of ways of giving each vault a type following the conditions modulo 10 9 \\u2009+\\u20097 .\\n\\n## Examples\\n\\n### Input #1\\n4 2\\n1 2\\n2 3\\n1 4\\n1 2\\n\\n### Output #1\\n1\\n\\n### Input #2\\n3 3\\n1 2\\n1 3\\n2 1\\n\\n### Output #2\\n13\\n\\n### Input #3\\n3 1\\n1 2\\n1 3\\n1 1\\n\\n### Output #3\\n0\\n\\n## Note\\n\\nIn test case 1 , we cannot have any vault of the highest security as its type is 1 implying that its adjacent vaults would have to have a vault type less than 1 , which is not allowed. Thus, there is only one possible combination, in which all the vaults have type 2 .\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the number of ways to assign types to $n$ vaults in a tree structure, such that certain conditions are met.\\nThe types are integers from 1 to $m$.\\nThere is a specific type $k$ which represents the highest security.\\nThere are constraints on the number of vaults of type $k$.\\nSpecifically:\\n1. There are at most $x$ vaults of type $k$.\\n2. If a vault is of type $k$, its adjacent vaults cannot be of type $k$.\\n3. If a vault is of type $k$, its adjacent vaults must have a type less than $k$.\\n4. All vaults have types between 1 and $m$.\\n\\nLet's break down the constraints.\\nThe vaults are nodes in a tree with $n$ nodes ($1 \\\\le n \\\\le 10^5$).\\nThe number of types $m$ can be up to $10^9$.\\nTh
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{"messages": ["{\"role\":\"user\",\"content\":\"You are a competitive programmer. You will be given a problem statement, please implement a solution in C++. The execution time and memory limit are also stated in the statement so be aware of the complexity of the program. Please wrap the code in ```cpp and ``` so that it is properly formatted. Your response should ONLY contain the C++ code, with no additional explanation or text.\\n\\nProblem:\\n\\nYou are solving a competitive programming problem.\\n\\n## Problem Statement\\n\\nYou've got another problem dealing with arrays. Let's consider an arbitrary sequence containing n (not necessarily different) integers a 1 , a 2 , ..., a n . We are interested in all possible pairs of numbers ( a i , a j ), ( 1\\u2009\\u2264\\u2009 i ,\\u2009 j \\u2009\\u2264\\u2009 n ). In other words, let's consider all n 2 pairs of numbers, picked from the given array. For example, in sequence a \\u2009=\\u2009{3,\\u20091,\\u20095} are 9 pairs of numbers: (3,\\u20093),\\u2009(3,\\u20091),\\u2009(3,\\u20095),\\u2009(1,\\u20093),\\u2009(1,\\u20091),\\u2009(1,\\u20095),\\u2009(5,\\u20093),\\u2009(5,\\u20091),\\u2009(5,\\u20095) . Let's sort all resulting pairs lexicographically by non-decreasing. Let us remind you that pair ( p 1 , q 1 ) is lexicographically less than pair ( p 2 , q 2 ) only if either p 1 < p 2 , or p 1 = p 2 and q 1 < q 2 . Then the sequence, mentioned above, will be sorted like that: (1,\\u20091),\\u2009(1,\\u20093),\\u2009(1,\\u20095),\\u2009(3,\\u20091),\\u2009(3,\\u20093),\\u2009(3,\\u20095),\\u2009(5,\\u20091),\\u2009(5,\\u20093),\\u2009(5,\\u20095) Let's number all the pair in the sorted list from 1 to n 2 . Your task is formulated like this: you should find the k -th pair in the ordered list of all possible pairs of the array you've been given.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 1 second\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains two integers n and k ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u200910 5 ,\\u20091\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u2009 n 2 ). The second line contains the array containing n integers a 1 , a 2 , ..., a n ( \\u2009-\\u200910 9 \\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u200910 9 ). The numbers in the array can coincide. All numbers are separated with spaces. Please do not use the %lld specificator to read or write 64-bit integers in \\u0421++. It is preferred to use cin , cout , streams or the %I64d specificator instead.\\n\\n## Output Specification\\n\\nIn the single line print two numbers \\u2014 the sought k -th pair.\\n\\n## Examples\\n\\n### Input #1\\n2 4\\n2 1\\n\\n### Output #1\\n2 2\\n\\n### Input #2\\n3 2\\n3 1 5\\n\\n### Output #2\\n1 3\\n\\n## Note\\n\\nIn the first sample the sorted sequence for the given array looks as: (1,\\u20091),\\u2009(1,\\u20092),\\u2009(2,\\u20091),\\u2009(2,\\u20092) . The 4 -th of them is pair (2,\\u20092) . The sorted sequence for the array from the second sample is given in the statement. The 2 -nd pair there is (1,\\u20093) .\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem asks us to find the $k$-th lexicographically smallest pair $(a_i, a_j)$ from a given array $a$ of size $n$. The pairs are formed by taking any two elements from the array (with replacement, i.e., $i$ and $j$ can be the same, and order matters). There are $n^2$ such pairs. The constraints are $n \\\\le 10^5$, and the values $a_i$ are between $-10^9$ and $10^9$. The time limit is 1 second.\\n\\nFirst, let's understand the lexicographical order. A pair $(p_1, q_1)$ is smaller than $(p_2, q_2)$ if $p_1 < p_2$ or ($p_1 = p_2$ and $q_1 < q_2$).\\nSince we are considering all pairs $(a_i, a_j)$, the first element of the pair comes from the set of values present in the array. If we sort the array $a$ in non-decreasing order, say $b_1, b_2, \\\\dots, b_n$, then the pairs will be generated based on the values in $b$.\\n\\nLet the sorted array
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