# 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.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.Examplesinput

??

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

Solution:

#include <bits/stdc++.h>

using namespace std;

template <typename T>
T inverse(T a, T m) {
T u = 0, v = 1;
while (a != 0) {
T t = m / a;
m -= t * a; swap(a, m);
u -= t * v; swap(u, v);
}
assert(m == 1);
return u;
}

template <typename T>
class Modular {
public:
using Type = typename decay<decltype(T::value)>::type;

constexpr Modular() : value() {}
template <typename U>
Modular(const U& x) {
value = normalize(x);
}

template <typename U>
static Type normalize(const U& x) {
Type v;
if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
else v = static_cast<Type>(x % mod());
if (v < 0) v += mod();
return v;
}

const Type& operator()() const { return value; }
template <typename U>
explicit operator U() const { return static_cast<U>(value); }
constexpr static Type mod() { return T::value; }

Modular& operator+=(const Modular& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
Modular& operator-=(const Modular& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
template <typename U> Modular& operator+=(const U& other) { return *this += Modular(other); }
template <typename U> Modular& operator-=(const U& other) { return *this -= Modular(other); }
Modular& operator++() { return *this += 1; }
Modular& operator--() { return *this -= 1; }
Modular operator++(int) { Modular result(*this); *this += 1; return result; }
Modular operator--(int) { Modular result(*this); *this -= 1; return result; }
Modular operator-() const { return Modular(-value); }

template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int>::value, Modular>::type& operator*=(const Modular& rhs) {
#ifdef _WIN32
uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;
asm(
"divl %4; \n\t"
: "=a" (d), "=d" (m)
: "d" (xh), "a" (xl), "r" (mod())
);
value = m;
#else
value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
return *this;
}
template <typename U = T>
typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type& operator*=(const Modular& rhs) {
int64_t q = static_cast<int64_t>(static_cast<long double>(value) * rhs.value / mod());
value = normalize(value * rhs.value - q * mod());
return *this;
}
template <typename U = T>
typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type& operator*=(const Modular& rhs) {
value = normalize(value * rhs.value);
return *this;
}

Modular& operator/=(const Modular& other) { return *this *= Modular(inverse(other.value, mod())); }

template <typename U>
friend const Modular<U>& abs(const Modular<U>& v) { return v; }

template <typename U>
friend bool operator==(const Modular<U>& lhs, const Modular<U>& rhs);

template <typename U>
friend bool operator<(const Modular<U>& lhs, const Modular<U>& rhs);

template <typename U>
friend std::istream& operator>>(std::istream& stream, Modular<U>& number);

private:
Type value;
};

template <typename T> bool operator==(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Modular<T>& lhs, U rhs) { return lhs == Modular<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) == rhs; }

template <typename T> bool operator!=(const Modular<T>& lhs, const Modular<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Modular<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Modular<T>& rhs) { return !(lhs == rhs); }

template <typename T> bool operator<(const Modular<T>& lhs, const Modular<T>& rhs) { return lhs.value < rhs.value; }

template <typename T> Modular<T> operator+(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template <typename T, typename U> Modular<T> operator+(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) += rhs; }

template <typename T> Modular<T> operator-(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template <typename T, typename U> Modular<T> operator-(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) -= rhs; }

template <typename T> Modular<T> operator*(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template <typename T, typename U> Modular<T> operator*(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) *= rhs; }

template <typename T> Modular<T> operator/(const Modular<T>& lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(const Modular<T>& lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template <typename T, typename U> Modular<T> operator/(U lhs, const Modular<T>& rhs) { return Modular<T>(lhs) /= rhs; }

template<typename T, typename U>
Modular<T> power(const Modular<T>& a, const U& b) {
assert(b >= 0);
Modular<T> x = a, res = 1;
U p = b;
while (p > 0) {
if (p & 1) res *= x;
x *= x;
p >>= 1;
}
return res;
}

template <typename T>
bool IsZero(const Modular<T>& number) {
return number() == 0;
}

template <typename T>
string to_string(const Modular<T>& number) {
}

template <typename T>
std::ostream& operator<<(std::ostream& stream, const Modular<T>& number) {
return stream << number();
}

template <typename T>
std::istream& operator>>(std::istream& stream, Modular<T>& number) {
typename common_type<typename Modular<T>::Type, int64_t>::type x;
stream >> x;
number.value = Modular<T>::normalize(x);
return stream;
}

/*
using ModType = int;

struct VarMod { static ModType value; };
ModType VarMod::value;
ModType& md = VarMod::value;
using Mint = Modular<VarMod>;
*/

constexpr int md = 998244353;
using Mint = Modular<std::integral_constant<decay<decltype(md)>::type, md>>;

int main() {
ios::sync_with_stdio(false);
cin.tie(0);
string s;
cin >> s;
int n = (int) s.size();
int l = 0, r = 0, q = 0;
for (char c : s) {
if (c == '(') ++l; else
if (c == ')') ++r;
else ++q;
}
vector<int> pref(n + 1);
for (int i = 0; i < n; i++) {
pref[i + 1] = pref[i] + (s[i] == '?');
}
vector<int> pref_open(n + 1);
for (int i = 0; i < n; i++) {
pref_open[i + 1] = pref_open[i] + (s[i] == '(');
}
vector<Mint> fact(n + 1);
fact[0] = 1;
for (int i = 1; i <= n; i++) {
fact[i] = fact[i - 1] * i;
}
vector<Mint> inv_fact(n + 1);
for (int i = 0; i <= n; i++) {
inv_fact[i] = 1 / fact[i];
}
auto C = [&](int N, int K) {
if (K < 0 || K > N) return Mint(0);
return fact[N] * inv_fact[K] * inv_fact[N - K];
};
Mint ans = 0;
for (int qtor = 0; qtor <= q; qtor++) {
int qtol = q - qtor;
int x = l + qtol;
int y = r + qtor;
int lft = pref[y];
int rgt = q - lft;
int z = pref_open[y];
ans += C(lft + rgt, qtol) * z;
ans += lft * C(lft + rgt - 1, qtol - 1);
}
cout << ans << '\n';
return 0;
}