Qwen3-4B-hardtests-frontier-filtered-full/val_dataset.jsonl

{"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 $\\\\gcd(i, p_i) > 1$ (call these \\\"good\\\" elements). If no such permutation exists, output -1.\\n\\nConstraints: $1 \\\\le n \\\\le 10^5$, $0 \\\\le k \\\\le n$.\\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 immediately means the maximum possible number of good elements is $n-1$. So if $k=n$, it's impossible.\\n\\nCheck with example 2: $n=1, k=1$. The only permutation is $[1]$, $\\\\gcd(1,1)=1$, so 0 good elements. We need $k=1$, impossible \\u2192 -1. This matches, and it's the $k=n$ case.\\n\\nFor $k < n$, we need a construction. The simplest way to make index $i$ good is $p_i = i$: then $\\\\gcd(i,i) = i > 1$ whenever $i \\\\ge 2$.\\n\\nSo the plan: pick $k$ indices from $\\\\{2,\\\\dots,n\\\\}$ to be good, and make the rest not good, while keeping it a valid permutation.\\n\\nLet's choose the last $k$ indices $\\\\{n-k+1, \\\\dots, n\\\\}$ as the good ones, setting $p_i = i$ there. The smallest index in this set is $n-k+1$. Since $k < n$ we have $n-k \\\\ge 1$, so $n-k+1 \\\\ge 2$. Thus every $i$ in this set is $\\\\ge 2$ and $\\\\gcd(i,i) = i > 1$ \\u2014 exactly $k$ good elements.\\n\\nNow the remaining indices $\\\\{1, \\\\dots, n-k\\\\}$ must be filled with values $\\\\{1, \\\\dots, n-k\\\\}$ so that none are good, i.e. $\\\\gcd(j, p_j) = 1$ for all of them. Let $m = n-k$. Use a cyclic shift: $p_j = j+1$ for $j < m$, and $p_m = 1$.\\n- For $j < m$: $\\\\gcd(j, j+1) = 1$. Always holds.\\n- For $j = m$: $\\\\gcd(m, 1) = 1$. Always holds.\\n\\nSo the first $m$ positions contribute 0 good elements, and the total is exactly 
{"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\\nJapate, while traveling through the forest of Mala, saw N bags of gold lying in a row. Each bag has some distinct weight of gold between 1 to N . Japate can carry only one bag of gold with him, so he uses the following strategy to choose a bag. Initially, he starts with an empty bag (zero weight). He considers the bags in some order. If the current bag has a higher weight than the bag in his hand, he picks the current bag. Japate put the bags in some order. Japate realizes that he will pick A bags, if he starts picking bags from the front, and will pick B bags, if he starts picking bags from the back. By picking we mean replacing the bag in his hand with the current one. Now he wonders how many permutations of bags are possible, in which he picks A bags from the front and B bags from back using the above strategy. Since the answer can be very large, output it modulo 998244353 .\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 3.5 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe only line of input contains three space separated integers N ( 1\\u2009\\u2264\\u2009 N \\u2009\\u2264\\u200910 5 ), A and B ( 0\\u2009\\u2264\\u2009 A ,\\u2009 B \\u2009\\u2264\\u2009 N ).\\n\\n## Output Specification\\n\\nOutput a single integer \\u2014 the number of valid permutations modulo 998244353 .\\n\\n## Examples\\n\\n### Input #1\\n1 1 1\\n\\n### Output #1\\n1\\n\\n### Input #2\\n2 1 1\\n\\n### Output #2\\n0\\n\\n### Input #3\\n2 2 1\\n\\n### Output #3\\n1\\n\\n### Input #4\\n5 2 2\\n\\n### Output #4\\n22\\n\\n## Note\\n\\nIn sample case 1 , the only possible permutation is [1] In sample cases 2 and 3 , only two permutations of size 2 are possible: {[1,\\u20092],\\u2009[2,\\u20091]} . The values of a and b for first permutation is 2 and 1 , and for the second permutation these values are 1 and 2 . In sample case 4 , out of 120 permutations of [1,\\u20092,\\u20093,\\u20094,\\u20095] possible, only 22 satisfy the given constraints of a and 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 us to find the number of permutations of length $N$ such that if we scan the permutation from the front, we pick $A$ bags, and if we scan from the back, we pick $B$ bags.\\nThe rule for picking a bag is: we start with a bag of weight 0. We iterate through the bags in the given order. If the current bag has a weight greater than the bag currently in hand, we pick it (replace the bag in hand).\\n\\nLet the permutation be $P = [p_1, p_2, \\\\dots, p_N]$. The weights are a permutation of $1, \\\\dots, N$.\\nScanning from front:\\nCurrent max weight $M = 0$.\\nCount $A = 0$.\\nFor $i = 1$ to $N$:\\n  If $p_i > M$:\\n    $M = p_i$\\n    $A = A + 1$\\n\\nScanning from back:\\nCurrent max weight $M = 0$.\\nCount $B = 0$.\\nFor $i = N$ down to $1$:\\n  If $p_i > M$:\\n    $M = p_i$\\n    $B = B + 1$\\n\\nWait, the problem statement says \\\"distinct weight of gold between 1 to N\\\". This is a standard permutation.\\nThe condition \\\"If the current bag has a higher weight than the bag in his hand, he picks the current bag\\\" describes the number of left-to-right maxima (or records) for the first count $A$, and right-to-left maxima for the second count $B$.\\n\\nLet's verify this interpretation.\\nSample 1: N=1, A=1, B=1. Permutation [1].\\nFront: 1 > 0 -> pick. Count = 1.\\nBack: 1 > 0 -> pick. Count = 1.\\nMatches.\\n\\nSample 2: N=2, A=1, B=1.\\nPermutations: [1, 
{"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 array of $n$ numbers $a_{1}, a_{2}, \\\\ldots, a_{n}$. Rearrange these numbers to satisfy $|a_{1} - a_{2}| \\\\le |a_{2} - a_{3}| \\\\le \\\\ldots \\\\le |a_{n-1} - a_{n}|$, where $|x|$ denotes absolute value of $x$. It's always possible to find such rearrangement. Note that all numbers in $a$ are not necessarily different. In other words, some numbers of $a$ may be same. You have to answer independent $t$ 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 a single integer $t$ ($1 \\\\le t \\\\le 10^{4}$)\\u00a0\\u2014 the number of test cases. The first line of each test case contains single integer $n$ ($3 \\\\le n \\\\le 10^{5}$)\\u00a0\\u2014 the length of array $a$. It is guaranteed that the sum of values of $n$ over all test cases in the input does not exceed $10^{5}$. The second line of each test case contains $n$ integers $a_{1}, a_{2}, \\\\ldots, a_{n}$ ($-10^{9} \\\\le a_{i} \\\\le 10^{9}$).\\n\\n## Output Specification\\n\\nFor each test case, print the rearranged version of array $a$ which satisfies given condition. If there are multiple valid rearrangements, print any of them.\\n\\n## Examples\\n\\n### Input #1\\n\\n2\\n6\\n5 -2 4 8 6 5\\n4\\n8 1 4 2\\n\\n\\n### Output #1\\n\\n5 5 4 6 8 -2\\n1 2 4 8\\n\\n## Note\\n\\nIn the first test case, after given rearrangement, $|a_{1} - a_{2}| = 0 \\\\le |a_{2} - a_{3}| = 1 \\\\le |a_{3} - a_{4}| = 2 \\\\le |a_{4} - a_{5}| = 2 \\\\le |a_{5} - a_{6}| = 10$. There are other possible answers like \\\" 5 4 5 6 -2 8 \\\". In the second test case, after given rearrangement, $|a_{1} - a_{2}| = 1 \\\\le |a_{2} - a_{3}| = 2 \\\\le |a_{3} - a_{4}| = 4$. There are other possible answers like \\\" 2 4 8 1 \\\".\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nWe need to rearrange the array so the absolute differences between adjacent elements are non-decreasing: $|a_1-a_2| \\\\le |a_2-a_3| \\\\le \\\\ldots$. We want small differences early, large ones late.\\n\\nSorting seems natural: let the sorted array be $s_0 \\\\le s_1 \\\\le \\\\ldots \\\\le s_{n-1}$, with gaps $g_i = s_{i+1} - s_i \\\\ge 0$. But simply printing the sorted array doesn't work \\u2014 e.g. $1, 10, 11, 20$ gives diffs $9, 1, 9$, not sorted. So we need a smarter order that still uses closeness in the sorted array.\\n\\nIdea: start from the middle of the sorted array and expand outwards, so that each new element straddles a wider range and the difference grows. Let me hypothesize the index order $m, m+1, m-1, m+2, m-2, \\\\dots$ where $m = (n-1)\\/2$.\\n\\nTest on example 1: input `5 -2 4 8 6 5`, sorted `-2 4 5 5 6 8` (indices 0..5), $m=2$. Order of indices $2,3,1,4,0,5$ \\u2192 values `5 5 4 6 -2 8`. Diffs: 0,1,2,8,10 \\u2014 non-decreasing, valid. (The sample output `5 5 4 6 8 -2` is a different valid arrangement; problem says print any.)\\n\\nTest on example 2: `8 1 4 2`, sorted `1 2 4 8`, $m=1$. Order $1,2,0,3$ \\u2192 `2 4 1 8`. Diffs 2,3,7 \\u2014 valid.\\n\\nNow let me prove this works in general. With the index sequence $m, m+1, m-1, m+2, m-2, \\\\dots$, the differences are:\\n- $|s_m - s_{m+1}| = g_m$\\n- $|s_{m+1} - s_{m-1}| = g_{m-1} + g_m$\\n- $|s_{m-1} - s_{m+2}| = g_{m-1} + g_m + g_{m+1}$\\n- $|s_{m+2} - s_{m-2}| = g_{m-2} + g_{m-1} + g_m + g_{m+1}$\\n- ...\\n\\nEach difference is a sum of consecutive gaps in the sorted array (this hol
{"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 forest \\u2014 an undirected graph with $n$ vertices such that each its connected component is a tree. The diameter (aka \\\"longest shortest path\\\") of a connected undirected graph is the maximum number of edges in the shortest path between any pair of its vertices. You task is to add some edges (possibly zero) to the graph so that it becomes a tree and the diameter of the tree is minimal possible. If there are multiple correct answers, print any of them.\\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 $m$ ($1 \\\\le n \\\\le 1000$, $0 \\\\le m \\\\le n - 1$) \\u2014 the number of vertices of the graph and the number of edges, respectively. Each of the next $m$ lines contains two integers $v$ and $u$ ($1 \\\\le v, u \\\\le n$, $v \\\\ne u$) \\u2014 the descriptions of the edges. It is guaranteed that the given graph is a forest.\\n\\n## Output Specification\\n\\nIn the first line print the diameter of the resulting tree. Each of the next $(n - 1) - m$ lines should contain two integers $v$ and $u$ ($1 \\\\le v, u \\\\le n$, $v \\\\ne u$) \\u2014 the descriptions of the added edges . The resulting graph should be a tree and its diameter should be minimal possible. For $m = n - 1$ no edges are added, thus the output consists of a single integer \\u2014 diameter of the given tree. If there are multiple correct answers, print any of them.\\n\\n## Examples\\n\\n### Input #1\\n\\n4 2\\n1 2\\n2 3\\n\\n\\n### Output #1\\n\\n2\\n4 2\\n\\n\\n### Input #2\\n\\n2 0\\n\\n\\n### Output #2\\n\\n1\\n1 2\\n\\n\\n### Input #3\\n\\n3 2\\n1 3\\n2 3\\n\\n\\n### Output #3\\n\\n2\\n\\n## Note\\n\\nIn the first example adding edges (1, 4) or (3, 4) will lead to a total diameter of 3. Adding edge (2, 4), however, will make it 2. Edge (1, 2) is the only option you have for the second example. The diameter is 1. You can't add any edges in the third example. The diameter is already 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: take a forest (collection of disjoint trees) and add edges to make it a single tree while minimizing the diameter. Output the minimum diameter and the added edges.\\n\\nLet the forest have $k$ components $T_1, \\\\dots, T_k$. We must add $k-1$ edges to connect them.\\n\\nLet $D_i$ be the diameter of $T_i$ and $R_i$ its radius. The radius is the minimum eccentricity over nodes, and for a tree $R_i = \\\\lceil D_i \\/ 2 \\\\rceil$. The center is the node achieving this minimum eccentricity, and it lies on the diameter path.\\n\\nWhen we connect two trees $T_i, T_j$ by an edge between $u \\\\in T_i$ and $v \\\\in T_j$, the longest path through the new edge is $dist(u, \\\\text{farthest in } T_i) + 1 + dist(v, \\\\text{farthest in } T_j)$. To minimize this, we should connect the *centers*, giving a path of $R_i + 1 + R_j$. So the new diameter is $\\\\max(D_i, D_j, R_i + 1 + R_j)$.\\n\\nNow for $k$ components. The natural idea: connect everything into a \\\"star of components\\\" with one hub component in the middle, connecting each other component's center to the hub's center. Which component should be the hub?\\n\\nLet me reason about the resulting diameter. Suppose we connect all $T_i$ ($i \\\\neq \\\\text{hub}$) to the hub at its center. The candidate longest paths are:\\n1. Entirely within some component: $D_i$.\\n2. Betwe
{"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\\nIgor likes hexadecimal notation and considers positive integer in the hexadecimal notation interesting if each digit and each letter in it appears no more than t times. For example, if t \\u2009=\\u20093 , then integers 13a13322 , aaa , abcdef0123456789 are interesting, but numbers aaaa , abababab and 1000000 are not interesting. Your task is to find the k -th smallest interesting for Igor integer in the hexadecimal notation. The integer should not contain leading zeros.\\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 the two integers k and t ( 1\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u20092\\u00b710 9 , 1\\u2009\\u2264\\u2009 t \\u2009\\u2264\\u200910 ) \\u2014 the number of the required integer and the maximum number of times some integer or letter can appear in interesting integer. It can be shown that the answer always exists for such constraints.\\n\\n## Output Specification\\n\\nPrint in the hexadecimal notation the only integer that is the k -th smallest interesting integer for Igor.\\n\\n## Examples\\n\\n### Input #1\\n17 1\\n\\n### Output #1\\n12\\n\\n### Input #2\\n1000000 2\\n\\n### Output #2\\nfca2c\\n\\n## Note\\n\\nThe first 20 interesting integers if t \\u2009=\\u20091 : 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , a , b , c , d , e , f , 10 , 12 , 13 , 14 , 15 . So the answer for the first example equals 12 .\\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 smallest \\\"interesting\\\" integer in hexadecimal notation, where each hex digit ($0..9, a..f$) appears at most $t$ times, with no leading zeros.\\nThe constraints are $k \\\\le 2 \\\\cdot 10^9$ and $t \\\\le 10$.\\n\\nSince $k$ is up to $2 \\\\cdot 10^9$, we cannot iterate through numbers. We need a way to count how many interesting numbers exist with a certain length or prefix, and then construct the $k$-th number digit by digit.\\n\\nIn base 16, a smaller length always implies a smaller value. So we first determine the length of the answer: check lengths $L=1, 2, 3, \\\\dots$ and sum up the counts of interesting numbers of length $L$ until the running total reaches $k$. Then within that length, we build the number digit by digit, greedily choosing the smallest digit that still leaves at least $k$ completions.\\n\\nHow large can the length get? $k$ is relatively small, so the length should be small.\\nLet's check the tightest case, $t=1$ (distinct digits only):\\n$L=1: 15$\\n$L=2: 15 \\\\times 15 = 225$\\n$L=3: 15 \\\\times 15 \\\\times 14 = 3150$\\n$L=4: 3150 \\\\times 13 = 40950$\\n$L=5: 40950 \\\\times 12 = 491400$\\n$L=6: 491400 \\\\times 11 = 5405400$\\n$L=7: 5405400 \\\\times 10 = 54054000$\\n$L=8: 54054000 \\\\times 9 = 486486000$\\nSum(1..8) $\\\\approx 5.46 \\\\cdot 10^8$, which is $< 2 \\\\cdot 10^9$.\\n$L=9: 486486000 \\\\times 8 = 3,891,888,000$.\\nSo for $k = 2 \\\\cdot 10^9$ with $t=1$, we need $L=9$.\\nFor larger $t$, counts grow faster (up to $\\\\sim 15 \\\\times 16^{L-1}$), so the length is smaller. E.g. for $t=10$, $16^8 \\\\approx 4.3 \\\\cdot 10^9 > 2 \\\\cdot 10^9$, so length is at most 8 or 9.\\nSo in all cases the answer length is at most about 9, and the maximum `len` we ever pass to a counting function is about 8. This bounds all counts by roughly $16^9 \\\\approx 6.8 \\\\cdot 10^{10}$, which fits comfortably in `long long`. (The problem guarantee
{"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\\nThe clique problem is one of the most well-known NP-complete problems. Under some simplification it can be formulated as follows. Consider an undirected graph G . It is required to find a subset of vertices C of the maximum size such that any two of them are connected by an edge in graph G . Sounds simple, doesn't it? Nobody yet knows an algorithm that finds a solution to this problem in polynomial time of the size of the graph. However, as with many other NP-complete problems, the clique problem is easier if you consider a specific type of a graph. Consider n distinct points on a line. Let the i -th point have the coordinate x i and weight w i . Let's form graph G , whose vertices are these points and edges connect exactly the pairs of points ( i ,\\u2009 j ) , such that the distance between them is not less than the sum of their weights, or more formally: | x i \\u2009-\\u2009 x j |\\u2009\\u2265\\u2009 w i \\u2009+\\u2009 w j . Find the size of the maximum clique in such graph.\\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 the integer n ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u2009200\\u2009000 ) \\u2014 the number of points. Each of the next n lines contains two numbers x i , w i ( 0\\u2009\\u2264\\u2009 x i \\u2009\\u2264\\u200910 9 ,\\u20091\\u2009\\u2264\\u2009 w i \\u2009\\u2264\\u200910 9 ) \\u2014 the coordinate and the weight of a point. All x i are different.\\n\\n## Output Specification\\n\\nPrint a single number \\u2014 the number of vertexes in the maximum clique of the given graph.\\n\\n## Examples\\n\\n### Input #1\\n4\\n2 3\\n3 1\\n6 1\\n0 2\\n\\n### Output #1\\n3\\n\\n## Note\\n\\nIf you happen to know how to solve this problem without using the specific properties of the graph formulated in the problem statement, then you are able to get a prize of one million dollars! The picture for the sample test.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nThe problem: find the maximum clique in a graph on $n$ points, where vertices $i,j$ are connected iff $|x_i - x_j| \\\\ge w_i + w_j$.\\n\\nLet me analyze the edge condition. WLOG assume $x_i < x_j$. Then $|x_i - x_j| \\\\ge w_i + w_j$ becomes $x_j - x_i \\\\ge w_i + w_j$, i.e. $x_j - w_j \\\\ge x_i + w_i$.\\n\\nThis suggests defining an interval for each point: $L_i = x_i - w_i$, $R_i = x_i + w_i$. Then the condition (for $x_i < x_j$) is $L_j \\\\ge R_i$ \\u2014 meaning interval $I_i$ lies completely to the left of $I_j$, i.e. the two intervals are disjoint (touching allowed).\\n\\nLet me prove edge $\\\\iff$ intervals disjoint in general. Suppose $I_i$ and $I_j$ overlap, so some point $y$ is in both: $x_i - w_i \\\\le y \\\\le x_i + w_i$ and $x_j - w_j \\\\le y \\\\le x_j + w_j$. Then $x_i - w_i \\\\le x_j + w_j$ and $x_j - w_j \\\\le x_i + w_i$, giving $|x_i - x_j| \\\\le w_i + w_j$. So the intervals overlap (positive-length intersection) exactly when there is NO edge. Conversely an edge exists iff the intervals are disjoint or merely touch at an endpoint (when $|x_i-x_j| = w_i+w_j$, equality still gives an edge). So touching counts as an edge \\u2014 that's the `>=` case.\\n\\nNow, a clique is a set of vertices where every pair is connected, i.e. every pair of intervals is pairwise disjoint. So the max clique size equals the maximum number of mutually disjoint intervals (touching allowed).\\n\\nOne thi
{"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\\nArkady plays Gardenscapes a lot. Arkady wants to build two new fountains. There are n available fountains, for each fountain its beauty and cost are known. There are two types of money in the game: coins and diamonds, so each fountain cost can be either in coins or diamonds. No money changes between the types are allowed. Help Arkady to find two fountains with maximum total beauty so that he can buy both at the same time.\\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 three integers n , c and d ( 2\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u2009100\\u2009000 , 0\\u2009\\u2264\\u2009 c ,\\u2009 d \\u2009\\u2264\\u2009100\\u2009000 )\\u00a0\\u2014 the number of fountains, the number of coins and diamonds Arkady has. The next n lines describe fountains. Each of these lines contain two integers b i and p i ( 1\\u2009\\u2264\\u2009 b i ,\\u2009 p i \\u2009\\u2264\\u2009100\\u2009000 )\\u00a0\\u2014 the beauty and the cost of the i -th fountain, and then a letter \\\" C \\\" or \\\" D \\\", describing in which type of money is the cost of fountain i : in coins or in diamonds, respectively.\\n\\n## Output Specification\\n\\nPrint the maximum total beauty of exactly two fountains Arkady can build. If he can't build two fountains, print 0 .\\n\\n## Examples\\n\\n### Input #1\\n3 7 6\\n10 8 C\\n4 3 C\\n5 6 D\\n\\n### Output #1\\n9\\n\\n### Input #2\\n2 4 5\\n2 5 C\\n2 1 D\\n\\n### Output #2\\n0\\n\\n### Input #3\\n3 10 10\\n5 5 C\\n5 5 C\\n10 11 D\\n\\n### Output #3\\n10\\n\\n## Note\\n\\nIn the first example Arkady should build the second fountain with beauty 4 , which costs 3 coins. The first fountain he can't build because he don't have enough coins. Also Arkady should build the third fountain with beauty 5 which costs 6 diamonds. Thus the total beauty of built fountains is 9 . In the second example there are two fountains, but Arkady can't build both of them, because he needs 5 coins for the first fountain, and Arkady has only 4 coins.\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nWe need to pick exactly two distinct fountains to maximize total beauty, subject to budget. Each fountain has beauty $b_i$, cost $p_i$, and a currency type C or D. Money doesn't convert between types, so the constraint depends on the pair of currencies.\\n\\nThere are three cases for a pair $(i, j)$:\\n1. Both in coins: $p_i + p_j \\\\le c$.\\n2. Both in diamonds: $p_i + p_j \\\\le d$.\\n3. One coin, one diamond: $p_i \\\\le c$ and $p_j \\\\le d$ (independent constraints).\\n\\nWe maximize $b_i + b_j$ over all valid distinct pairs.\\n\\nConstraints: $n, c, d \\\\le 100{,}000$, $b_i, p_i \\\\le 100{,}000$. Time limit 2s. This allows $O(n \\\\log n)$ or $O(n + \\\\max p)$ \\u2014 since costs are bounded by $10^5$, an array indexed by cost is feasible.\\n\\nSeparate fountains into a coins list and a diamonds list, then handle the three scenarios.\\n\\n**Scenario 3** is easiest. Just take the max beauty among coin fountains with $p \\\\le c$, call it $max\\\\_b\\\\_c$, and the max beauty among diamond fountains with $p \\\\le d$, call it $max\\\\_b\\\\_d$. If both exist (both $> 0$, valid since $b \\\\ge 1$), the candidate is their sum. They're automatically distinct fountains since one is C and one is D, so no distinctness check needed.\\n\\n**Scenarios 1 and 2** are the same problem: given items with weig
{"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 an undirected unweighted connected graph consisting of $n$ vertices and $m$ edges. It is guaranteed that there are no self-loops or multiple edges in the given graph. Your task is to choose at most $\\\\lfloor\\\\frac{n}{2}\\\\rfloor$ vertices in this graph so each unchosen vertex is adjacent (in other words, connected by an edge) to at least one of chosen vertices. It is guaranteed that the answer exists. If there are multiple answers, you can print any. You will be given multiple independent queries to answer.\\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 $t$ ($1 \\\\le t \\\\le 2 \\\\cdot 10^5$) \\u2014 the number of queries. Then $t$ queries follow. The first line of each query contains two integers $n$ and $m$ ($2 \\\\le n \\\\le 2 \\\\cdot 10^5$, $n - 1 \\\\le m \\\\le min(2 \\\\cdot 10^5, \\\\frac{n(n-1)}{2})$) \\u2014 the number of vertices and the number of edges, respectively. The following $m$ lines denote edges: edge $i$ is represented by a pair of integers $v_i$, $u_i$ ($1 \\\\le v_i, u_i \\\\le n$, $u_i \\\\ne v_i$), which are the indices of vertices connected by the edge. There are no self-loops or multiple edges in the given graph, i.\\u2009e. for each pair ($v_i, u_i$) there are no other pairs ($v_i, u_i$) or ($u_i, v_i$) in the list of edges, and for each pair ($v_i, u_i$) the condition $v_i \\\\ne u_i$ is satisfied. It is guaranteed that the given graph is connected . It is guaranteed that $\\\\sum m \\\\le 2 \\\\cdot 10^5$ over all queries.\\n\\n## Output Specification\\n\\nFor each query print two lines. In the first line print $k$ ($1 \\\\le \\\\lfloor\\\\frac{n}{2}\\\\rfloor$) \\u2014 the number of chosen vertices. In the second line print $k$ distinct integers $c_1, c_2, \\\\dots, c_k$ in any order, where $c_i$ is the index of the $i$-th chosen vertex. It is guaranteed that the answer exists. If there are multiple answers, you can print any.\\n\\n## Examples\\n\\n### Input #1\\n\\n2\\n4 6\\n1 2\\n1 3\\n1 4\\n2 3\\n2 4\\n3 4\\n6 8\\n2 5\\n5 4\\n4 3\\n4 1\\n1 3\\n2 3\\n2 6\\n5 6\\n\\n\\n### Output #1\\n\\n2\\n1 3\\n3\\n4 3 6\\n\\n## Note\\n\\nIn the first query any vertex or any pair of vertices will suffice. Note that you don't have to minimize the number of chosen vertices. In the second query two vertices can be enough (vertices $2$ and $4$) but three is also ok.\\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 a set of at most $\\\\lfloor n\\/2 \\\\rfloor$ vertices such that every unchosen vertex is adjacent to at least one chosen vertex. That is exactly a dominating set of size $\\\\le n\\/2$. The graph is connected and an answer is guaranteed.\\n\\nThe condition is: for $S$ the chosen set, every $v \\\\in V \\\\setminus S$ has a neighbor in $S$. We need $|S| \\\\le \\\\lfloor n\\/2 \\\\rfloor$.\\n\\nKey observation: consider a spanning tree of the graph. A solution for the spanning tree is also a solution for the graph, since tree edges are a subset of graph edges. A tree is bipartite, so we can 2-color it into sets $A$ and $B$ (even\\/odd depth in a BFS tree).\\n\\nClaim: both $A$ and $B$ are dominating sets. Take $A$ as the chosen set. Every $v \\\\in B$ has a parent in the BFS tree which lies in $A$, so $v$ has a neighbor in $A$. Symmetrically, every $v \\\\in A$ that is not the root has a p
{"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\\nIt is the middle of 2018 and Maria Stepanovna, who lives outside Krasnokamensk (a town in Zabaikalsky region), wants to rent three displays to highlight an important problem. There are $n$ displays placed along a road, and the $i$-th of them can display a text with font size $s_i$ only. Maria Stepanovna wants to rent such three displays with indices $i < j < k$ that the font size increases if you move along the road in a particular direction. Namely, the condition $s_i < s_j < s_k$ should be held. The rent cost is for the $i$-th display is $c_i$. Please determine the smallest cost Maria Stepanovna should pay.\\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 $n$ ($3 \\\\le n \\\\le 3\\\\,000$)\\u00a0\\u2014 the number of displays. The second line contains $n$ integers $s_1, s_2, \\\\ldots, s_n$ ($1 \\\\le s_i \\\\le 10^9$)\\u00a0\\u2014 the font sizes on the displays in the order they stand along the road. The third line contains $n$ integers $c_1, c_2, \\\\ldots, c_n$ ($1 \\\\le c_i \\\\le 10^8$)\\u00a0\\u2014 the rent costs for each display.\\n\\n## Output Specification\\n\\nIf there are no three displays that satisfy the criteria, print -1 . Otherwise print a single integer\\u00a0\\u2014 the minimum total rent cost of three displays with indices $i < j < k$ such that $s_i < s_j < s_k$.\\n\\n## Examples\\n\\n### Input #1\\n5\\n2 4 5 4 10\\n40 30 20 10 40\\n\\n### Output #1\\n90\\n\\n### Input #2\\n3\\n100 101 100\\n2 4 5\\n\\n### Output #2\\n-1\\n\\n### Input #3\\n10\\n1 2 3 4 5 6 7 8 9 10\\n10 13 11 14 15 12 13 13 18 13\\n\\n### Output #3\\n33\\n\\n## Note\\n\\nIn the first example you can, for example, choose displays $1$, $4$ and $5$, because $s_1 < s_4 < s_5$ ($2 < 4 < 10$), and the rent cost is $40 + 10 + 40 = 90$. In the second example you can't select a valid triple of indices, so the answer is -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 minimum cost to rent three displays with indices $i < j < k$ such that $s_i < s_j < s_k$. We minimize $c_i + c_j + c_k$, or output -1 if no such triple exists.\\n\\nThe constraints are $n \\\\le 3000$, $s_i$ up to $10^9$, $c_i$ up to $10^8$.\\nSince $n \\\\le 3000$, an $O(n^2)$ solution is acceptable. An $O(n^3)$ solution would be too slow ($3000^3 = 2.7 \\\\times 10^{10}$).\\n\\nKey idea: iterate over the middle element $j$. Since $i < j < k$, $j$ ranges from $2$ to $n-1$ (1-based).\\nFor a fixed $j$, we need the index $i < j$ with $s_i < s_j$ and $c_i$ minimized \\u2014 call it `min_c_i`. Similarly the index $k > j$ with $s_k > s_j$ and $c_k$ minimized \\u2014 call it `min_c_k`.\\nIf both exist, the best cost with this $j$ as middle is `min_c_i` + $c_j$ + `min_c_k`. Take the global minimum over all valid $j$. If for some $j$ no valid $i$ or $k$ exists, that $j$ cannot be a middle element.\\n\\nAlgorithm (0-based indexing in code, $j$ from $1$ to $n-2$):\\n1. Initialize `ans` to -1 (sentinel meaning \\\"not found yet\\\"; safe since all costs are $\\\\ge 1$).\\n2. For each $j$: scan $i$ from $0$ to $j-1$, and whenever $s_i < s_j$, track the minimum $c_i$. If none found, skip $j$.\\n3. Then scan $k$ from $j+1$ to $n-1$, and whenever $s_k > s_j$, track the minimum $c_k$. If none found, skip $j$.\\n4. Compute the sum and update `ans` (using the `ans == -1 || current < ans` check 
{"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 string $s$, consisting of lowercase Latin letters. While there is at least one character in the string $s$ that is repeated at least twice , you perform the following operation:  you choose the index $i$ ($1 \\\\le i \\\\le |s|$) such that the character at position $i$ occurs at least two times in the string $s$, and delete the character at position $i$, that is, replace $s$ with $s_1 s_2 \\\\ldots s_{i-1} s_{i+1} s_{i+2} \\\\ldots s_n$. For example, if $s=$ \\\"codeforces\\\" , then you can apply the following sequence of operations:  $i=6 \\\\Rightarrow s=$ \\\"codefrces\\\" ;  $i=1 \\\\Rightarrow s=$ \\\"odefrces\\\" ;  $i=7 \\\\Rightarrow s=$ \\\"odefrcs\\\" ; Given a given string $s$, find the lexicographically maximum string that can be obtained after applying a certain sequence of operations after which all characters in the string become unique . A string $a$ of length $n$ is lexicographically less than a string $b$ of length $m$, if:  there is an index $i$ ($1 \\\\le i \\\\le \\\\min(n, m)$) such that the first $i-1$ characters of the strings $a$ and $b$ are the same, and the $i$-th character of the string $a$ is less than $i$-th character of string $b$; or the first $\\\\min(n, m)$ characters in the strings $a$ and $b$ are the same and $n < m$. For example, the string $a=$ \\\"aezakmi\\\" is lexicographically less than the string $b=$ \\\"aezus\\\" .\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 2.5 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains one integer $t$ ($1 \\\\le t \\\\le 10^4$). Then $t$ test cases follow. Each test case is characterized by a string $s$, consisting of lowercase Latin letters ($1 \\\\le |s| \\\\le 2 \\\\cdot 10^5$). It is guaranteed that the sum of the lengths of the strings in all test cases does not exceed $2 \\\\cdot 10^5$.\\n\\n## Output Specification\\n\\nFor each test case, output the lexicographically maximum string that can be obtained after applying a certain sequence of operations after which all characters in the string become unique .\\n\\n## Examples\\n\\n### Input #1\\n\\n6\\ncodeforces\\naezakmi\\nabacaba\\nconvexhull\\nswflldjgpaxs\\nmyneeocktxpqjpz\\n\\n\\n### Output #1\\n\\nodfrces\\nezakmi\\ncba\\nconvexhul\\nwfldjgpaxs\\nmyneocktxqjpz\\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: repeatedly delete characters that appear at least twice, stopping when all characters are unique, and maximize the result lexicographically.\\n\\nLet me analyze the operation. We can delete an instance of a character only while it occurs at least twice. So if a character appears once, it can never be deleted \\u2014 we can't delete its last instance, because to delete it, it must appear \\u22652 times. This means every character that appears in $s$ must survive exactly once in the final string. The process ends when no character repeats, so the final string contains exactly one copy of each distinct character of $s$.\\n\\nAlso, we only delete, never reorder, so the relative order of the kept characters is preserved. Therefore the problem is equivalent to: select a subsequence of $s$ containing exactly one occurrence of each distinct character, lexicographically maximal.\\n\\nThis is exactly the \\\"Remove Duplicate Letters\\\" problem (LeetCode 316), but inverted \\u2014 we want the lexicographically **largest** such subsequence instead
{"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\\nToday at the lesson Vitya learned a very interesting function\\u00a0\\u2014 mex . Mex of a sequence of numbers is the minimum non-negative number that is not present in the sequence as element. For example, mex ([4,\\u200933,\\u20090,\\u20091,\\u20091,\\u20095])\\u2009=\\u20092 and mex ([1,\\u20092,\\u20093])\\u2009=\\u20090 . Vitya quickly understood all tasks of the teacher, but can you do the same? You are given an array consisting of n non-negative integers, and m queries. Each query is characterized by one number x and consists of the following consecutive steps: Perform the bitwise addition operation modulo 2 ( xor ) of each array element with the number x .  Find mex of the resulting array. Note that after each query the array changes.\\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 integer numbers n and m ( 1\\u2009\\u2264\\u2009 n ,\\u2009 m \\u2009\\u2264\\u20093\\u00b710 5 )\\u00a0\\u2014 number of elements in array and number of queries. Next line contains n integer numbers a i ( 0\\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u20093\\u00b710 5 )\\u00a0\\u2014 elements of then array. Each of next m lines contains query\\u00a0\\u2014 one integer number x ( 0\\u2009\\u2264\\u2009 x \\u2009\\u2264\\u20093\\u00b710 5 ).\\n\\n## Output Specification\\n\\nFor each query print the answer on a separate line.\\n\\n## Examples\\n\\n### Input #1\\n2 2\\n1 3\\n1\\n3\\n\\n### Output #1\\n1\\n0\\n\\n### Input #2\\n4 3\\n0 1 5 6\\n1\\n2\\n4\\n\\n### Output #2\\n2\\n0\\n0\\n\\n### Input #3\\n5 4\\n0 1 5 6 7\\n1\\n1\\n4\\n5\\n\\n### Output #3\\n2\\n2\\n0\\n2\\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>\\nWe need MEX of the array after XORing all elements with a cumulative value. The XOR is persistent, so let $S$ be the cumulative XOR of all previous queries (initially 0). For a query $x$, the effective mask becomes $K = S \\\\oplus x$, and we then update $S \\\\leftarrow K$. We want $\\\\text{mex}(\\\\{a_i \\\\oplus K\\\\})$.\\n\\nConstraints: $n, m \\\\le 3 \\\\cdot 10^5$, $a_i, x \\\\le 3 \\\\cdot 10^5$. Since $x < 2^{19}$, the cumulative XOR $K$ also stays $< 2^{19}$.\\n\\nKey reduction: we want the smallest $v \\\\ge 0$ not present in $\\\\{a_i \\\\oplus K\\\\}$. But $v = a_i \\\\oplus K$ for some $i$ iff $v \\\\oplus K = a_i \\\\in A$. So we want the smallest $v \\\\ge 0$ such that $v \\\\oplus K \\\\notin A$, where $A$ is the original set. Let $T = v \\\\oplus K$ be the \\\"target\\\" value we look up in $A$.\\n\\nSince $a_i \\\\le 3 \\\\cdot 10^5 < 2^{19}$, and $n < 2^{19}$, the MEX can never reach $2^{19}$ (that would require all $2^{19}$ values present). So working with about 19\\u201320 bits suffices. I'll use 20 bits to be safe.\\n\\nApproach: build a binary trie over the distinct values of $A$, MSB first. Each node stores `cnt` = number of distinct values in its subtree, and two child pointers. A subtree at bit level $b$ (covering the remaining $b$ bits below it) has capacity $2^b$; it is \\\"full\\\" (contains every value with that prefix) iff `cnt == 2^b`.\\n\\n**Why distinct values matter:** if we inserted duplicates, `cnt` would be inflated and the `cnt == 2^b` test would misfire \\u2014 e.g. $\\\\{0,0,0\\\\}$ would give a leaf `cnt` of 3 while capacity is 1, so `cnt == 1` fails and we'd wrongly think 0 is missing. So I sort and `unique` the ar
{"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\\nDaniel has a string s , consisting of lowercase English letters and period signs (characters ' . '). Let's define the operation of replacement as the following sequence of steps: find a substring \\\" .. \\\" (two consecutive periods) in string s , of all occurrences of the substring let's choose the first one, and replace this substring with string \\\" . \\\". In other words, during the replacement operation, the first two consecutive periods are replaced by one. If string s contains no two consecutive periods, then nothing happens. Let's define f ( s ) as the minimum number of operations of replacement to perform, so that the string does not have any two consecutive periods left. You need to process m queries, the i -th results in that the character at position x i ( 1\\u2009\\u2264\\u2009 x i \\u2009\\u2264\\u2009 n ) of string s is assigned value c i . After each operation you have to calculate and output the value of f ( s ) . Help Daniel to process all queries.\\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\\u2009\\u2264\\u2009 n ,\\u2009 m \\u2009\\u2264\\u2009300\\u2009000 ) the length of the string and the number of queries. The second line contains string s , consisting of n lowercase English letters and period signs. The following m lines contain the descriptions of queries. The i -th line contains integer x i and c i ( 1\\u2009\\u2264\\u2009 x i \\u2009\\u2264\\u2009 n , c i \\u2014 a lowercas English letter or a period sign), describing the query of assigning symbol c i to position x i .\\n\\n## Output Specification\\n\\nPrint m numbers, one per line, the i -th of these numbers must be equal to the value of f ( s ) after performing the i -th assignment.\\n\\n## Examples\\n\\n### Input #1\\n10 3\\n.b..bz....\\n1 h\\n3 c\\n9 f\\n\\n### Output #1\\n4\\n3\\n1\\n\\n### Input #2\\n4 4\\n.cc.\\n2 .\\n3 .\\n2 a\\n1 a\\n\\n### Output #2\\n1\\n3\\n1\\n1\\n\\n## Note\\n\\nNote to the first sample test (replaced periods are enclosed in square brackets). The original string is \\\" .b..bz.... \\\". after the first query f ( hb..bz.... ) = 4\\u00a0\\u00a0\\u00a0\\u00a0(\\\" hb[..]bz.... \\\" \\u2009\\u2192\\u2009 \\\" hb.bz[..].. \\\" \\u2009\\u2192\\u2009 \\\" hb.bz[..]. \\\" \\u2009\\u2192\\u2009 \\\" hb.bz[..] \\\" \\u2009\\u2192\\u2009 \\\" hb.bz. \\\") after the second query f ( hb\\u0441.bz.... ) = 3\\u00a0\\u00a0\\u00a0\\u00a0(\\\" hb\\u0441.bz[..].. \\\" \\u2009\\u2192\\u2009 \\\" hb\\u0441.bz[..]. \\\" \\u2009\\u2192\\u2009 \\\" hb\\u0441.bz[..] \\\" \\u2009\\u2192\\u2009 \\\" hb\\u0441.bz. \\\") after the third query f ( hb\\u0441.bz..f. ) = 1\\u00a0\\u00a0\\u00a0\\u00a0(\\\" hb\\u0441.bz[..]f. \\\" \\u2009\\u2192\\u2009 \\\" hb\\u0441.bz.f. \\\") Note to the second sample test. The original string is \\\" .cc. \\\". after the first query: f ( ..c. ) = 1\\u00a0\\u00a0\\u00a0\\u00a0(\\\" [..]c. \\\" \\u2009\\u2192\\u2009 \\\" .c. \\\") after the second query: f ( .... ) = 3\\u00a0\\u00a0\\u00a0\\u00a0(\\\" [..].. \\\" \\u2009\\u2192\\u2009 \\\" [..]. \\\" \\u2009\\u2192\\u2009 \\\" [..] \\\" \\u2009\\u2192\\u2009 \\\" . \\\") after the third query: f ( .a.. ) = 1\\u00a0\\u00a0\\u00a0\\u00a0(\\\" .a[..] \\\" \\u2009\\u2192\\u2009 \\\" .a. \\\") after the fourth query: f ( aa.. ) = 1\\u00a0\\u00a0\\u00a0\\u00a0(\\\" aa[..] \\\" \\u2009\\u2192\\u2009 \\\" aa. \\\")\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenera
{"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\\nSo the Beautiful Regional Contest (BeRC) has come to an end! $n$ students took part in the contest. The final standings are already known: the participant in the $i$-th place solved $p_i$ problems. Since the participants are primarily sorted by the number of solved problems, then $p_1 \\\\ge p_2 \\\\ge \\\\dots \\\\ge p_n$. Help the jury distribute the gold, silver and bronze medals. Let their numbers be $g$, $s$ and $b$, respectively. Here is a list of requirements from the rules, which all must be satisfied: for each of the three types of medals, at least one medal must be awarded (that is, $g>0$, $s>0$ and $b>0$);  the number of gold medals must be strictly less than the number of silver and the number of bronze (that is, $g<s$ and $g<b$, but there are no requirements between $s$ and $b$);  each gold medalist must solve strictly more problems than any awarded with a silver medal;  each silver medalist must solve strictly more problems than any awarded a bronze medal;  each bronze medalist must solve strictly more problems than any participant not awarded a medal;  the total number of medalists $g+s+b$ should not exceed half of all participants (for example, if $n=21$, then you can award a maximum of $10$ participants, and if $n=26$, then you can award a maximum of $13$ participants). The jury wants to reward with medals the total maximal number participants (i.e. to maximize $g+s+b$) so that all of the items listed above are fulfilled. Help the jury find such a way to award medals.\\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 the input contains an integer $t$ ($1 \\\\le t \\\\le 10000$) \\u2014 the number of test cases in the input. Then $t$ test cases follow. The first line of a test case contains an integer $n$ ($1 \\\\le n \\\\le 4\\\\cdot10^5$) \\u2014 the number of BeRC participants. The second line of a test case contains integers $p_1, p_2, \\\\dots, p_n$ ($0 \\\\le p_i \\\\le 10^6$), where $p_i$ is equal to the number of problems solved by the $i$-th participant from the final standings. The values $p_i$ are sorted in non-increasing order, i.e. $p_1 \\\\ge p_2 \\\\ge \\\\dots \\\\ge p_n$. The sum of $n$ over all test cases in the input does not exceed $4\\\\cdot10^5$.\\n\\n## Output Specification\\n\\nPrint $t$ lines, the $j$-th line should contain the answer to the $j$-th test case. The answer consists of three non-negative integers $g, s, b$. Print $g=s=b=0$ if there is no way to reward participants with medals so that all requirements from the statement are satisfied at the same time.  Otherwise, print three positive numbers $g, s, b$ \\u2014 the possible number of gold, silver and bronze medals, respectively. The sum of $g+s+b$ should be the maximum possible. If there are several answers, print any of them.\\n\\n## Examples\\n\\n### Input #1\\n\\n5\\n12\\n5 4 4 3 2 2 1 1 1 1 1 1\\n4\\n4 3 2 1\\n1\\n1000000\\n20\\n20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1\\n32\\n64 64 63 58 58 58 58 58 37 37 37 37 34 34 28 28 28 28 28 28 24 24 19 17 17 17 17 16 16 16 16 11\\n\\n\\n### Output #1\\n\\n1 2 3\\n0 0 0\\n0 0 0\\n2 5 3\\n2 6 6\\n\\n## Note\\n\\nIn the first test case, it is possible to reward $1$ gold, $2$ silver and $3$ bronze medals. In this case, the participant solved $5$ tasks will be rewarded with the gold medal, participants solved $4$ tasks will be rewarded with silver medals, participants solved $2$ or $3$ tasks will be rewarded with bronze medals. Participants solved exactl
{"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 been given n distinct integers a 1 ,\\u2009 a 2 ,\\u2009...,\\u2009 a n . You can remove at most k of them. Find the minimum modular m ( m \\u2009>\\u20090) , so that for every pair of the remaining integers ( a i ,\\u2009 a j ) , the following unequality holds: .\\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 k ( 1\\u2009\\u2009\\u2264\\u2009 n \\u2009\\u2009\\u2264\\u20095000,\\u20090\\u2009\\u2264\\u2009 k \\u2009\\u2264\\u20094 ), which we have mentioned above. The second line contains n distinct integers a 1 ,\\u2009 a 2 ,\\u2009...,\\u2009 a n (0\\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u200910 6 ) .\\n\\n## Output Specification\\n\\nPrint a single positive integer \\u2014 the minimum m .\\n\\n## Examples\\n\\n### Input #1\\n7 0\\n0 2 3 6 7 12 18\\n\\n### Output #1\\n13\\n\\n### Input #2\\n7 1\\n0 2 3 6 7 12 18\\n\\n### Output #2\\n7\\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 find the minimum positive integer $m$ such that we can remove at most $k$ integers from a given set of $n$ distinct integers $a_1, a_2, \\\\dots, a_n$, and the remaining integers are all distinct modulo $m$. That is, for any two remaining integers $x, y$, $x \\\\not\\\\equiv y \\\\pmod m$.\\n\\nThe constraints are:\\n$n \\\\le 5000$\\n$k \\\\le 4$\\n$a_i \\\\le 10^6$\\nTime limit: 2 seconds\\nMemory limit: 256 MB\\n\\nLet $S$ be the set of remaining integers. The condition is that for all $x, y \\\\in S$ with $x \\\\neq y$, $x \\\\not\\\\equiv y \\\\pmod m$. This is equivalent to saying that for all distinct $x, y \\\\in S$, $m$ does not divide $|x - y|$.\\nIf we want to check if a specific $m$ is valid, we need to see if there exists a subset of size at least $n - k$ such that no two elements have a difference divisible by $m$.\\n\\nSince $k$ is very small ($k \\\\le 4$), this suggests that we are looking for a modulus $m$ that \\\"conflicts\\\" with very few pairs of numbers.\\nSpecifically, if we fix $m$, a pair $(a_i, a_j)$ is a \\\"bad pair\\\" if $a_i \\\\equiv a_j \\\\pmod m$.\\nThe condition \\\"remove at most $k$ elements\\\" means that in the graph where vertices are the original numbers and edges connect pairs $(a_i, a_j)$ such that $a_i \\\\equiv a_j \\\\pmod m$, we can remove at most $k$ vertices to eliminate all edges. This is the Vertex Cover problem.\\nHowever, the structure of the graph is specific. The edges are defined by congruence modulo $m$.\\nActually, if we group the numbers by their remainder modulo $m$, say $R_r = \\\\{a_i \\\\mid a_i \\\\equiv r \\\\pmod m\\\\}$, then within each group $R_r$, every pair of elements has a difference divisible by $m$. To satisfy the condition, we can keep at most one element from each group $R_r$.\\nWait, the problem says \\\"remove at most $k$\\\". If we keep a subset $S$, then for every pair in $S$, their difference is not divisible by $m$. This means no two elements in $S$ can have the same remainder modulo $m$.\\nSo, if we partition the original set $A = \\\\{a_1, \\\\dots, a_n\\\\}$ into sets $R_0, R_1, \\\\dots, R_{m-1}$ based on $a_i \\\\pmod m$, then from each $R_r$, we can select at most one element to be in $S$.\\nThe number of elements we keep is $|S|$. The number of elements we remove is $n - |S|$. We require $n - |S| \\\\le k$, or $|S| \\\\ge n - 
{"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 statistic of price changes for one product represented as an array of $n$ positive integers $p_0, p_1, \\\\dots, p_{n - 1}$, where $p_0$ is the initial price of the product and $p_i$ is how the price was increased during the $i$-th month. Using these price changes you are asked to calculate the inflation coefficients for each month as the ratio of current price increase $p_i$ to the price at the start of this month $(p_0 + p_1 + \\\\dots + p_{i - 1})$. Your boss said you clearly that the inflation coefficients must not exceed $k$ %, so you decided to increase some values $p_i$ in such a way, that all $p_i$ remain integers and the inflation coefficients for each month don't exceed $k$ %. You know, that the bigger changes\\u00a0\\u2014 the more obvious cheating. That's why you need to minimize the total sum of changes. What's the minimum total sum of changes you need to make all inflation coefficients not more than $k$ %?\\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 $t$ ($1 \\\\le t \\\\le 1000$)\\u00a0\\u2014 the number of test cases. The first line of each test case contains two integers $n$ and $k$ ($2 \\\\le n \\\\le 100$; $1 \\\\le k \\\\le 100$)\\u00a0\\u2014 the length of array $p$ and coefficient $k$. The second line of each test case contains $n$ integers $p_0, p_1, \\\\dots, p_{n - 1}$ ($1 \\\\le p_i \\\\le 10^9$)\\u00a0\\u2014 the array $p$.\\n\\n## Output Specification\\n\\nFor each test case, print the minimum total sum of changes you need to make all inflation coefficients not more than $k$ %.\\n\\n## Examples\\n\\n### Input #1\\n\\n2\\n4 1\\n20100 1 202 202\\n3 100\\n1 1 1\\n\\n\\n### Output #1\\n\\n99\\n0\\n\\n## Note\\n\\nIn the first test case, you can, for example, increase $p_0$ by $50$ and $p_1$ by $49$ and get array $[20150, 50, 202, 202]$. Then you get the next inflation coefficients:  $\\\\frac{50}{20150} \\\\le \\\\frac{1}{100}$;  $\\\\frac{202}{20150 + 50} \\\\le \\\\frac{1}{100}$;  $\\\\frac{202}{20200 + 202} \\\\le \\\\frac{1}{100}$; In the second test case, you don't need to modify array $p$, since the inflation coefficients are already good:  $\\\\frac{1}{1} \\\\le \\\\frac{100}{100}$;  $\\\\frac{1}{1 + 1} \\\\le \\\\frac{100}{100}$;\\n\\nYour final response should only contain the C++ solution, wrapped in ```cpp and ```.\\n\\nGenerate solution code:\"}", "{\"role\":\"assistant\",\"content\":\"<think>\\nWe need to minimize the total sum of increases to the array $p$ such that for all $i \\\\ge 1$, $\\\\frac{p_i}{\\\\sum_{j=0}^{i-1} p_j} \\\\le \\\\frac{k}{100}$. We can increase any $p_i$ by non-negative integers.\\n\\nLet $S_{i-1} = \\\\sum_{j=0}^{i-1} p'_j$ be the prefix sum. The condition rewrites as $100 \\\\cdot p'_i \\\\le k \\\\cdot S_{i-1}$, i.e. $S_{i-1} \\\\ge \\\\frac{100 p'_i}{k}$.\\n\\nConstraints: $n \\\\le 100$, $k \\\\le 100$, $p_i \\\\le 10^9$. So an $O(n)$ per test case works easily; we just need `long long` for the big values.\\n\\nKey question: when a condition is violated, what do we increase? The condition at $i$ is $p'_i \\\\le \\\\frac{k}{100} S_{i-1}$. Increasing $p'_i$ raises the numerator, making the condition *harder* \\u2014 so we never increase $p'_i$ to fix its own condition. We must increase the prefix sum $S_{i-1}$, i.e. add to some $p'_j$ with $j < i$.\\n\\nWhich $j$? Compare adding $\\\\Delta$ to $p_0$ vs to $p_{i-1}$:\\n- Both raise $S_{i-1}$ by $\\\\Delta$, and both raise every future denominator $S_{j-
{"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\\nEvery year Santa Claus gives gifts to all children. However, each country has its own traditions, and this process takes place in different ways. For example, in Berland you need to solve the New Year's puzzle. Polycarp got the following problem: given a grid strip of size $2 \\\\times n$, some cells of it are blocked. You need to check if it is possible to tile all free cells using the $2 \\\\times 1$ and $1 \\\\times 2$ tiles (dominoes). For example, if $n = 5$ and the strip looks like this (black cells are blocked): Then it can be tiled, for example, using two vertical and two horizontal tiles, as in the picture below (different tiles are marked by different colors). And if $n = 3$ and the strip looks like this: It is impossible to tile free cells. Polycarp easily solved this task and received his New Year's gift. Can you solve it?\\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 $t$ ($1 \\\\leq t \\\\leq 10^4$)\\u00a0\\u2014 the number of test cases. Then $t$ test cases follow. Each test case is preceded by an empty line. The first line of each test case contains two integers $n$ and $m$ ($1 \\\\le n \\\\le 10^9$, $1 \\\\le m \\\\le 2 \\\\cdot 10^5$)\\u00a0\\u2014 the length of the strip and the number of blocked cells on it. Each of the next $m$ lines contains two integers $r_i, c_i$ ($1 \\\\le r_i \\\\le 2, 1 \\\\le c_i \\\\le n$)\\u00a0\\u2014 numbers of rows and columns of blocked cells. It is guaranteed that all blocked cells are different, i.e. $(r_i, c_i) \\\\ne (r_j, c_j), i \\\\ne j$. It is guaranteed that the sum of $m$ over all test cases does not exceed $2 \\\\cdot 10^5$.\\n\\n## Output Specification\\n\\nFor each test case, print on a separate line:  \\\" YES \\\", if it is possible to tile all unblocked squares with the $2 \\\\times 1$ and $1 \\\\times 2$ tiles;  \\\" NO \\\" otherwise. You can output \\\" YES \\\" and \\\" NO \\\" in any case (for example, the strings yEs , yes , Yes and YES will be recognized as positive).\\n\\n## Examples\\n\\n### Input #1\\n\\n3\\n\\n5 2\\n2 2\\n1 4\\n\\n3 2\\n2 1\\n2 3\\n\\n6 4\\n2 1\\n2 3\\n2 4\\n2 6\\n\\n\\n### Output #1\\n\\nYES\\nNO\\nNO\\n\\n## Note\\n\\nThe first two test cases are explained in the statement. In the third test case the strip looks like this:  It is easy to check that the unblocked squares on it can not be tiled.\\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 a $2 \\\\times n$ grid with some blocked cells can be completely tiled by $2 \\\\times 1$ and $1 \\\\times 2$ dominoes. The blocked cells are given as coordinates $(r, c)$.\\n\\nThe grid has 2 rows and $n$ columns. $n$ can be up to $10^9$, so we cannot simulate the grid directly. However, the number of blocked cells $m$ is up to $2 \\\\cdot 10^5$. This suggests that we should focus on the blocked cells and the gaps between them.\\n\\nLet's represent the blocked cells. We can store them in a set or a sorted vector of pairs $(r, c)$. Since the grid is only 2 rows high, we can think of this as a sequence of columns.\\nActually, since $n$ is large, we can't iterate through columns. But the blocked cells partition the columns into segments.\\nWait, a blocked cell at column $c$ affects the tiling around column $c$.\\nIf there are no blocked cells, the condition is simply that the total number of free cells must 
{"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\\nIt is only a few days until Seollal (Korean Lunar New Year), and Jaehyun has invited his family to his garden. There are kids among the guests. To make the gathering more fun for the kids, Jaehyun is going to run a game of hide-and-seek. The garden can be represented by a $n \\\\times m$ grid of unit cells. Some (possibly zero) cells are blocked by rocks, and the remaining cells are free. Two cells are neighbors if they share an edge. Each cell has up to 4 neighbors: two in the horizontal direction and two in the vertical direction. Since the garden is represented as a grid, we can classify the cells in the garden as either \\\" black \\\" or \\\" white \\\". The top-left cell is black, and two cells which are neighbors must be different colors. Cell indices are 1-based, so the top-left corner of the garden is cell $(1, 1)$. Jaehyun wants to turn his garden into a maze by placing some walls between two cells. Walls can only be placed between neighboring cells. If the wall is placed between two neighboring cells $a$ and $b$, then the two cells $a$ and $b$ are not neighboring from that point. One can walk directly between two neighboring cells if and only if there is no wall directly between them. A maze must have the following property. For each pair of free cells in the maze, there must be exactly one simple path between them. A simple path between cells $a$ and $b$ is a sequence of free cells in which the first cell is $a$, the last cell is $b$, all cells are distinct, and any two consecutive cells are neighbors which are not directly blocked by a wall. At first, kids will gather in cell $(1, 1)$, and start the hide-and-seek game. A kid can hide in a cell if and only if that cell is free, it is not $(1, 1)$, and has exactly one free neighbor. Jaehyun planted roses in the black cells, so it's dangerous if the kids hide there. So Jaehyun wants to create a maze where the kids can only hide in white cells. You are given the map of the garden as input. Your task is to help Jaehyun create a maze.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 3 seconds\\nMemory Limit: 1024 megabytes\\n\\n## Input Specification\\n\\nYour program will be judged in multiple test cases. The first line contains the number of test cases $t$. ($1 \\\\le t \\\\le 100$). Afterward, $t$ test cases with the described format will be given. The first line of a test contains two integers $n, m$ ($2 \\\\le n, m \\\\le 20$), the size of the grid. In the next $n$ line of a test contains a string of length $m$, consisting of the following characters (without any whitespace): O : A free cell. X : A rock. It is guaranteed that the first cell (cell $(1, 1)$) is free, and every free cell is reachable from $(1, 1)$. If $t \\\\geq 2$ is satisfied, then the size of the grid will satisfy $n \\\\le 10, m \\\\le 10$ . In other words, if any grid with size $n > 10$ or $m > 10$ is given as an input, then it will be the only input on the test case ($t = 1$).\\n\\n## Output Specification\\n\\nFor each test case, print the following: If there are no possible mazes, print a single line NO . Otherwise, print a single line YES , followed by a grid of size $(2n-1) \\\\times (2m-1)$ denoting the found maze. The rules for displaying the maze follows. All cells are indexed in 1-base. For all $1 \\\\le i \\\\le n, 1 \\\\le j \\\\le m$, if the cell $(i, j)$ is free cell, print 'O' in the cell $(2i-1, 2j-1)$. Otherwise, print 'X' in the cell $(2i-1, 2j-1)$.  For all $1 \\\\le i \\\\le n, 1 \\\\le j \\\\le m-1$, if the neighboring cell $(i, j), (i, j+1)$ 
{"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\\nThe Berland Kingdom is a set of n cities connected with each other with n \\u2009-\\u20091 railways. Each road connects exactly two different cities. The capital is located in city 1 . For each city there is a way to get from there to the capital by rail. In the i -th city there is a soldier division number i , each division is characterized by a number of a i . It represents the priority, the smaller the number, the higher the priority of this division. All values of a i are different. One day the Berland King Berl Great declared a general mobilization, and for that, each division should arrive in the capital. Every day from every city except the capital a train departs. So there are exactly n \\u2009-\\u20091 departing trains each day. Each train moves toward the capital and finishes movement on the opposite endpoint of the railway on the next day. It has some finite capacity of c j , expressed in the maximum number of divisions, which this train can transport in one go. Each train moves in the direction of reducing the distance to the capital. So each train passes exactly one railway moving from a city to the neighboring (where it stops) toward the capital. In the first place among the divisions that are in the city, division with the smallest number of a i get on the train, then with the next smallest and so on, until either the train is full or all the divisions are be loaded. So it is possible for a division to stay in a city for a several days. The duration of train's progress from one city to another is always equal to 1 day. All divisions start moving at the same time and end up in the capital, from where they don't go anywhere else any more. Each division moves along a simple path from its city to the capital, regardless of how much time this journey will take. Your goal is to find for each division, in how many days it will arrive to the capital of Berland. The countdown begins from day 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 the single integer n ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u20095000 ). It is the number of cities in Berland. The second line contains n space-separated integers a 1 ,\\u2009 a 2 ,\\u2009...,\\u2009 a n , where a i represents the priority of the division, located in the city number i . All numbers a 1 ,\\u2009 a 2 ,\\u2009...,\\u2009 a n are different ( 1\\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u200910 9 ). Then n \\u2009-\\u20091 lines contain the descriptions of the railway roads. Each description consists of three integers v j ,\\u2009 u j ,\\u2009 c j , where v j , u j are number of cities connected by the j -th rail, and c j stands for the maximum capacity of a train riding on this road ( 1\\u2009\\u2264\\u2009 v j ,\\u2009 u j \\u2009\\u2264\\u2009 n ,\\u2009 v j \\u2009\\u2260\\u2009 u j , 1\\u2009\\u2264\\u2009 c j \\u2009\\u2264\\u2009 n ).\\n\\n## Output Specification\\n\\nPrint sequence t 1 ,\\u2009 t 2 ,\\u2009...,\\u2009 t n , where t i stands for the number of days it takes for the division of city i to arrive to the capital. Separate numbers with spaces.\\n\\n## Examples\\n\\n### Input #1\\n4\\n40 10 30 20\\n1 2 1\\n2 3 1\\n4 2 1\\n\\n### Output #1\\n0 1 3 2 \\n\\n### Input #2\\n5\\n5 4 3 2 1\\n1 2 1\\n2 3 1\\n2 4 1\\n4 5 1\\n\\n### Output #2\\n0 1 4 2 3\\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\",\"conte
{"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 an array a consisting of n integers. Each integer from 1 to n appears exactly once in this array. For some indices i ( 1\\u2009\\u2264\\u2009 i \\u2009\\u2264\\u2009 n \\u2009-\\u20091 ) it is possible to swap i -th element with ( i \\u2009+\\u20091) -th, for other indices it is not possible. You may perform any number of swapping operations any order. There is no limit on the number of times you swap i -th element with ( i \\u2009+\\u20091) -th (if the position is not forbidden). Can you make this array sorted in ascending order performing some sequence of swapping operations?\\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 n ( 2\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u2009200000 ) \\u2014 the number of elements in the array. The second line contains n integers a 1 , a 2 , ..., a n ( 1\\u2009\\u2264\\u2009 a i \\u2009\\u2264\\u2009200000 ) \\u2014 the elements of the array. Each integer from 1 to n appears exactly once. The third line contains a string of n \\u2009-\\u20091 characters, each character is either 0 or 1 . If i -th character is 1 , then you can swap i -th element with ( i \\u2009+\\u20091) -th any number of times, otherwise it is forbidden to swap i -th element with ( i \\u2009+\\u20091) -th.\\n\\n## Output Specification\\n\\nIf it is possible to sort the array in ascending order using any sequence of swaps you are allowed to make, print YES . Otherwise, print NO .\\n\\n## Examples\\n\\n### Input #1\\n6\\n1 2 5 3 4 6\\n01110\\n\\n### Output #1\\nYES\\n\\n### Input #2\\n6\\n1 2 5 3 4 6\\n01010\\n\\n### Output #2\\nNO\\n\\n## Note\\n\\nIn the first example you may swap a 3 and a 4 , and then swap a 4 and a 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 whether we can sort a permutation of $1 \\\\dots n$ given a set of allowed adjacent swaps. We have array $a$ of size $n$ and string $s$ of length $n-1$. If $s[i] == '1'$, we can swap $a[i]$ and $a[i+1]$ (0-based for implementation). Any number of swaps.\\n\\nLet's analyze the structure. The allowed swaps define edges between indices: an edge between $i$ and $i+1$ if $s[i] == '1'$. Since edges only exist between adjacent indices, this graph is a collection of disjoint paths \\u2014 contiguous segments. Let these connected components of indices be $C_1, \\\\dots, C_k$.\\n\\nKey property: within a component, adjacent swaps allow generating any permutation of the values currently at those positions (bubble sort within the range). But no value can cross a component boundary \\u2014 to move a value from position $u$ to $v$, there must be a path of allowed swaps between them, so $u$ and $v$ must be in the same component.\\n\\nSo the condition for sortability is: for every component $[L, R]$, the set of values currently there, $\\\\{a[L], \\\\dots, a[R]\\\\}$, must equal the target set $\\\\{L+1, \\\\dots, R+1\\\\}$ (0-based indices, 1-based values; sorted array has $a[i]=i+1$).\\n\\nWhy necessary and sufficient: within a component we can sort locally, so if the value set matches the target set, sorting the subarray puts everything right. Conversely, if some value $v \\\\in \\\\{L+1,\\\\dots,R+1\\\\}$ sits at a position $p \\\\notin [L,R]$, it must reach position $v-1 \\\\in [L,R]$, but it can't cross the boundary \\u2014 impossible.\\n\\nNow, how to check this efficiently. I coul
{"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\\nIahub isn't well prepared on geometry problems, but he heard that this year there will be a lot of geometry problems on the IOI selection camp. Scared, Iahub locked himself in the basement and started thinking of new problems of this kind. One of them is the following. Iahub wants to draw n distinct points and m segments on the OX axis. He can draw each point with either red or blue. The drawing is good if and only if the following requirement is met: for each segment [ l i ,\\u2009 r i ] consider all the red points belong to it ( r i points), and all the blue points belong to it ( b i points); each segment i should satisfy the inequality | r i \\u2009-\\u2009 b i |\\u2009\\u2264\\u20091 . Iahub thinks that point x belongs to segment [ l ,\\u2009 r ] , if inequality l \\u2009\\u2264\\u2009 x \\u2009\\u2264\\u2009 r holds. Iahub gives to you all coordinates of points and segments. Please, help him to find any good drawing.\\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 ( 1\\u2009\\u2264\\u2009 n \\u2009\\u2264\\u2009100 ) and m ( 1\\u2009\\u2264\\u2009 m \\u2009\\u2264\\u2009100 ). The next line contains n space-separated integers x 1 ,\\u2009 x 2 ,\\u2009...,\\u2009 x n ( 0\\u2009\\u2264\\u2009 x i \\u2009\\u2264\\u2009100 ) \\u2014 the coordinates of the points. The following m lines contain the descriptions of the m segments. Each line contains two integers l i and r i ( 0\\u2009\\u2264\\u2009 l i \\u2009\\u2264\\u2009 r i \\u2009\\u2264\\u2009100 ) \\u2014 the borders of the i -th segment. It's guaranteed that all the points are distinct.\\n\\n## Output Specification\\n\\nIf there is no good drawing for a given test, output a single integer -1. Otherwise output n integers, each integer must be 0 or 1. The i -th number denotes the color of the i -th point (0 is red, and 1 is blue). If there are multiple good drawings you can output any of them.\\n\\n## Examples\\n\\n### Input #1\\n3 3\\n3 7 14\\n1 5\\n6 10\\n11 15\\n\\n### Output #1\\n0 0 0\\n\\n### Input #2\\n3 4\\n1 2 3\\n1 2\\n2 3\\n5 6\\n2 2\\n\\n### Output #2\\n1 0 1\\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>\\nWe need to color each of $n$ distinct points 0 (red) or 1 (blue) so that for every segment $[l_i, r_i]$, the number of red $r_i$ and blue $b_i$ points inside satisfy $|r_i - b_i| \\\\le 1$. Output any valid coloring, or -1.\\n\\nConstraints are tiny: $n, m \\\\le 100$, coordinates $\\\\le 100$, all points distinct.\\n\\nLet me reformulate. If red = 0, blue = 1, then in a segment with $|S_k|$ points, $|r_i - b_i| \\\\le 1$ means the count of each color differs by at most one \\u2014 the coloring must be almost balanced on every segment.\\n\\nKey observation: the points lie on a line. Sort them by coordinate: $X_1 < X_2 < \\\\dots < X_n$. Any segment $[l,r]$ contains exactly the points with coordinate in $[l,r]$, and since the points are sorted this is always a *contiguous* subsegment $\\\\{P_L,\\\\dots,P_R\\\\}$. Proof: if $X_a < X_b < X_c$ and $X_a, X_c \\\\in [l,r]$, then $l \\\\le X_a < X_b < X_c \\\\le r$, so $X_b \\\\in [l,r]$ too. So no segment ever selects a \\\"gap\\\".\\n\\nThat reduces the whole problem to: color a sequence so that every contiguous subsegment is color-balanced within 1. An alternating coloring $0,1,0,1,\\\\dots$ on the sorted points d
{"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\\nIn this problem you will have to deal with a very special network. The network consists of two parts: part A and part B . Each part consists of n vertices; i -th vertex of part A is denoted as A i , and i -th vertex of part B is denoted as B i . For each index i ( 1\\u2009\\u2264\\u2009 i \\u2009<\\u2009 n ) there is a directed edge from vertex A i to vertex A i \\u2009+\\u20091 , and from B i to B i \\u2009+\\u20091 , respectively. Capacities of these edges are given in the input. Also there might be several directed edges going from part A to part B (but never from B to A ). You have to calculate the maximum flow value from A 1 to B n in this network. Capacities of edges connecting A i to A i \\u2009+\\u20091 might sometimes change, and you also have to maintain the maximum flow value after these changes. Apart from that, the network is fixed (there are no changes in part B , no changes of edges going from A to B , and no edge insertions or deletions). Take a look at the example and the notes to understand the structure of the network better.\\n\\n## Time Limit and Memory Limit\\n\\nTime Limit: 4 seconds\\nMemory Limit: 256 megabytes\\n\\n## Input Specification\\n\\nThe first line contains three integer numbers n , m and q ( 2\\u2009\\u2264\\u2009 n ,\\u2009 m \\u2009\\u2264\\u20092\\u00b710 5 , 0\\u2009\\u2264\\u2009 q \\u2009\\u2264\\u20092\\u00b710 5 ) \\u2014 the number of vertices in each part, the number of edges going from A to B and the number of changes, respectively. Then n \\u2009-\\u20091 lines follow, i -th line contains two integers x i and y i denoting that the edge from A i to A i \\u2009+\\u20091 has capacity x i and the edge from B i to B i \\u2009+\\u20091 has capacity y i ( 1\\u2009\\u2264\\u2009 x i ,\\u2009 y i \\u2009\\u2264\\u200910 9 ). Then m lines follow, describing the edges from A to B . Each line contains three integers x , y and z denoting an edge from A x to B y with capacity z ( 1\\u2009\\u2264\\u2009 x ,\\u2009 y \\u2009\\u2264\\u2009 n , 1\\u2009\\u2264\\u2009 z \\u2009\\u2264\\u200910 9 ). There might be multiple edges from A x to B y . And then q lines follow, describing a sequence of changes to the network. i -th line contains two integers v i and w i , denoting that the capacity of the edge from A v i to A v i \\u2009+\\u20091 is set to w i ( 1\\u2009\\u2264\\u2009 v i \\u2009<\\u2009 n , 1\\u2009\\u2264\\u2009 w i \\u2009\\u2264\\u200910 9 ).\\n\\n## Output Specification\\n\\nFirstly, print the maximum flow value in the original network. Then print q integers, i -th of them must be equal to the maximum flow value after i -th change.\\n\\n## Examples\\n\\n### Input #1\\n4 3 2\\n1 2\\n3 4\\n5 6\\n2 2 7\\n1 4 8\\n4 3 9\\n1 100\\n2 100\\n\\n### Output #1\\n9\\n14\\n14\\n\\n## Note\\n\\nThis is the original network in the example:\\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 flow from $A_1$ to $B_n$ in a specific type of network and handle updates to the capacities of edges within part A ($A_i \\\\to A_{i+1}$).\\n\\nThe network structure is:\\n- Two chains of vertices: $A_1 \\\\to A_2 \\\\to \\\\dots \\\\to A_n$ and $B_1 \\\\to B_2 \\\\to \\\\dots \\\\to B_n$.\\n- Edges $A_i \\\\to A_{i+1}$ have capacity $x_i$.\\n- Edges $B_i \\\\to B_{i+1}$ have capacity $y_i$.\\n- There are edges from some $A_u$ to some $B_v$ with capacity $z$.\\n- Source is $A_1$, sink is $B_n$.\\n\\nWe need to output the max