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D2. Beautiful Bracket Sequence (hard version)
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

This is the hard version of this problem. The only difference is the limit of $$$n$$$ - the length of the input string. In this version, $$$1 \leq n \leq 10^6$$$.

Let's define a correct bracket sequence and its depth as follow:

  • An empty string is a correct bracket sequence with depth $$$0$$$.
  • If "s" is a correct bracket sequence with depth $$$d$$$ then "(s)" is a correct bracket sequence with depth $$$d + 1$$$.
  • If "s" and "t" are both correct bracket sequences then their concatenation "st" is a correct bracket sequence with depth equal to the maximum depth of $$$s$$$ and $$$t$$$.

For a (not necessarily correct) bracket sequence $$$s$$$, we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from $$$s$$$ (possibly zero). For example: the bracket sequence $$$s = $$$"())(())" has depth $$$2$$$, because by removing the third character we obtain a correct bracket sequence "()(())" with depth $$$2$$$.

Given a string $$$a$$$ consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in $$$a$$$ by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo $$$998244353$$$.

Hacks in this problem can be done only if easy and hard versions of this problem was solved.

Input

The only line contains a non-empty string consist of only '(', ')' and '?'. The length of the string is at most $$$10^6$$$.

Output

Print the answer modulo $$$998244353$$$ in a single line.

Examples
Input
??
Output
1
Input
(?(?))
Output
9
Note

In the first test case, we can obtain $$$4$$$ bracket sequences by replacing all characters '?' with either '(' or ')':

  • "((". Its depth is $$$0$$$;
  • "))". Its depth is $$$0$$$;
  • ")(". Its depth is $$$0$$$;
  • "()". Its depth is $$$1$$$.

So, the answer is $$$1 = 0 + 0 + 0 + 1$$$.

In the second test case, we can obtain $$$4$$$ bracket sequences by replacing all characters '?' with either '(' or ')':

  • "(((())". Its depth is $$$2$$$;
  • "()()))". Its depth is $$$2$$$;
  • "((()))". Its depth is $$$3$$$;
  • "()(())". Its depth is $$$2$$$.

So, the answer is $$$9 = 2 + 2 + 3 + 2$$$.