Beautiful Bracket Sequence (hard version)

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.