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ambitious

Find the longest valid slice of a sequence of brackets after performing at most K bracket rotations.

Programming language:

A bracket sequence is considered to be a valid bracket expression if any of the following conditions is true:

- it is empty;
- it has the form "
(U)" whereUis a valid bracket sequence;- it has the form "
VW" whereVandWare valid bracket sequences.

For example, the sequence "`(())()`" is a valid bracket expression, but "`((())(()`" is not.

You are given a sequence of brackets S and you are allowed to rotate some of them. Bracket rotation means picking a single bracket and changing it into its opposite form (i.e. an opening bracket can be changed into a closing bracket and vice versa). The goal is to find the longest slice (contiguous substring) of S that forms a valid bracket sequence using at most K bracket rotations.

Write a function:

class Solution { public int solution(String S, int K); }

that, given a string S consisting of N brackets and an integer K, returns the length of the maximum slice of S that can be transformed into a valid bracket sequence by performing at most K bracket rotations.

For example, given S = "`)()()(`" and K = 3, you can rotate the first and last brackets to get "`(()())`", which is a valid bracket sequence, so the function should return 6 (notice that you need to perform only two rotations in this instance, though).

Given S = "`)))(((`" and K = 2, you can rotate the second and fifth brackets to get "`)()()(`", which has a substring "`()()`" that is a valid bracket sequence, so the function should return 4.

Given S = "`)))(((`" and K = 0, you can't rotate any brackets, and since there is no valid bracket sequence with a positive length in string S, the function should return 0.

Write an ** efficient** algorithm for the following assumptions:

- string S contains only brackets: '
(' or ')';- N is an integer within the range [1..30,000];
- K is an integer within the range [0..N].

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