One day, Wet Shark was given an array . As always, he started playing with its subsequences.When you came to know about this habit, you presented him a task of finding all pairs of subsequences, , which satisfies all of the following constraints. We will represent a pair of subsequence as and and must be of same length, i.e., .Please help Wet Shark determine how many possible subsequences and can exist. Because the number of choices may be big, output your answer modulo .Note:Two segments are different if there's exists at least one index such that element is present in exactly one of them.Both subsequences can overlap each other.Subsequences do not necessarily have to be distinctInput FormatThe first line consists of 3 space-separated integers , , , where denotes the length of the original array, , and and are as defined above.The next line contains space-separated integers, , representing the elements of .ConstraintsOutput FormatOutput total number of pairs of subsequences, , satisfying the above conditions. As the number can be large, output it's modulo

You are given an integer array nums and a non-negative integer k. A sequence of integers seq is called good if there are at most k indices i in the range [0, seq.length - 2] such that seq[i] != seq[i + 1].Return the maximum possible length of a good subsequence of nums. Example 1:Input: nums = [1,2,1,1,3], k = 2Output: 4Explanation:The maximum length subsequence is [1,2,1,1,3].Example 2:Input: nums = [1,2,3,4,5,1], k = 0Output: 2Explanation:The maximum length subsequence is [1,2,3,4,5,1]. Constraints:1 <= nums.length <= 5001 <= nums[i] <= 1090 <= k <= min(nums.length, 25)Python3 1class Solution:2    def maximumLength(self, nums: List[int], k: int) -> int:

Given a string s of length n, find all the possible subsequences of the string s in lexicographically-sorted order.

Nikita is a student passionate about number theory and algorithms. He faces an interesting problem related to an array of numbers.Suppose Nikita has an array of integers a𝑎 of length n𝑛. He will call a subsequence†† of the array special if its least common multiple (LCM) is not contained in a𝑎. The LCM of an empty subsequence is equal to 00.Nikita wonders: what is the length of the longest special subsequence of a𝑎? Help him answer this question!†† A sequence b𝑏 is a subsequence of a𝑎 if b𝑏 can be obtained from a𝑎 by the deletion of several (possibly, zero or all) elements, without changing the order of the remaining elements. For example, [5,2,3][5,2,3] is a subsequence of [1,5,7,8,2,4,3][1,5,7,8,2,4,3].InputEach test contains multiple test cases. The first line of input contains a single integer t𝑡 (1≤t≤20001≤𝑡≤2000) — the number of test cases. The description of the test cases follows.The first line of each test case contains a single integer n𝑛 (1≤n≤20001≤𝑛≤2000) — the length of the array a𝑎.The second line of each test case contains n𝑛 integers a1,a2,…,an𝑎1,𝑎2,…,𝑎𝑛 (1≤ai≤1091≤𝑎𝑖≤109) — the elements of the array a𝑎.It is guaranteed that the sum of n𝑛 over all test cases does not exceed 20002000.OutputFor each test case, output a single integer — the length of the longest special subsequence of a𝑎.ExampleinputCopy651 2 4 8 1663 2 10 20 60 172 3 4 6 12 100003 120003692 42 7 3 6 7 7 1 684 99 57 179 10203 2 11 4081211outputCopy044580NoteIn the first test case, the LCM of any non-empty subsequence is contained in a𝑎, so the answer is 00.In the second test case, we can take the subsequence [3,2,10,1][3,2,10,1], its LCM is equal to 3030, which is not contained in a𝑎.In the third test case, we can take the subsequence [2,3,6,100003][2,3,6,100003], its LCM is equal to 600018600018, which is not contained in a𝑎.

Given an array, find the length of the longest subsequence whose elements can be re-arranged in a strictly increasing contiguous order. The difference between 2 adjacent elements in the subsequence, after re-arrangement, should be exactly 1.Input FormatThe first line of input contains T - the number of test cases. It's followed by 2T lines. The first line of each test case contains N - size of the array. The next line contains N integers - the elements of the array.Output FormatFor each test case, print the length of the longest subsequence, separated by a new line.Constraints1 <= T <= 10001 <= N <= 10000-100000 <= ar[i] <= 100000ExampleInput3821 -22 -22 5 -31 -24 5 -231018 -33 31 33 30 -14 32 30 16 1766 3 8 5 2 5Output342ExplanationTest Case 1Subsequence is: -22, -24, -23.Test Case 2Subsequence is: 31, 33, 30, 32.Test Case 3Subsequence is: 6, 5 or 3, 2.

You are given an integer array nums.A subsequence sub of nums with length x is called valid if it satisfies:(sub[0] + sub[1]) % 2 == (sub[1] + sub[2]) % 2 == ... == (sub[x - 2] + sub[x - 1]) % 2.Return the length of the longest valid subsequence of nums.A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements. Example 1:Input: nums = [1,2,3,4]Output: 4Explanation:The longest valid subsequence is [1, 2, 3, 4].Example 2:Input: nums = [1,2,1,1,2,1,2]Output: 6Explanation:The longest valid subsequence is [1, 2, 1, 2, 1, 2].Example 3:Input: nums = [1,3]Output: 2Explanation:The longest valid subsequence is [1, 3]. Constraints:2 <= nums.length <= 2 * 1051 <= nums[i] <= 107