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 $9 = 2 + 2 + 3 + 2$.