Beautiful Bracket Sequence (hard version)

给定一个长度为$n$的字符串，其中只有```(, ), ?```三种字符，其中```?```可以为```(```或者```)``` 对于一个括号序列，定义其权值为其通过**删除字符**后可以得到的合法的括号匹配的最深的深度，下面是一些括号匹配的例子： 深度为$1:()$ 深度为$2:(())$ 深度为$3:((()))$ 下面这个例子是一个深度为$0$的括号序列，当然其权值也为$0$ 深度为$0:(($ 您希望求出所有可能的括号序列（即问号替换后）的权值，答案对$998244353$取模 翻译：by Soulist

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.

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

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

??

1

(?(?))

9

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 $ .