#pragma comment(linker, "/STACK: 268435456")
#pragma GCC optimization ("O3")
#pragma GCC optimization ("unroll-loops")
#pragma GCC optimize("Ofast")
#pragma GCC target("avx,avx2,fma")
#define _CRT_SECURE_NO_WARNINGS
#include <bits/stdc++.h>
#include <unordered_map>
#include <unordered_set>
#include <chrono>
using namespace std;
using namespace chrono;
#define IOS ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define int long long
#define double long double
#define sqr(x) (x)*(x)
#define pb push_back
typedef int C;
typedef complex<C>P;
#define X real()
#define Y imag()
#define ins insert
#define amas(a) for (auto&c:a) cout<<c<<" ";
#define pf push_front
#define pii pair<int,int>
struct custom_hash {
	static uint64_t splitmix64(uint64_t x) {
		x += 0x9e3779b97f4a7c15;
		x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
		x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
		return x ^ (x >> 31);
	}
	size_t operator()(uint64_t x) const {
		static const uint64_t fixed_random = steady_clock::now().time_since_epoch().count();
		return splitmix64(x + fixed_random);
	}
};
//void bdt(vector<vector<int>>& g, int x, int lx, int rx) { //[l;r)
//	if (rx - lx == 1) {
//		return;
//	}
//	int m = (lx + rx) / 2;
//	g[x].push_back(2 * x + 1);
//	g[2 * x + 1].push_back(x);
//	bdt(g, 2 * x + 1, lx, m);
//	g[x].push_back(2 * x + 2);
//	g[2 * x + 2].push_back(x);
//	bdt(g, 2 * x + 2, m, rx);
//}
vector<int>rans;
void ans(int x, vector<char>& val) {
	switch (val[x])
	{ 
	case '0': rans[x] = rans[2 * x + 2]; break;
	case '1': rans[x] = rans[2 * x + 1]; break;
	case '?': rans[x] = rans[2 * x + 2] + rans[2 * x + 1]; break;
	}
	if (x == 0) return;
	ans((x - 1) / 2, val);
}
void fcnt(int x, vector<char>& val,int lx,int rx) {
	if (rx - lx == 1) {
		rans[x] = 1;
		return;
	}
	int m = (lx + rx) / 2;
	fcnt(2 * x + 1, val, lx, m);
	fcnt(2 * x + 2, val, m, rx);
	switch (val[x])
	{
	case '0': rans[x] = rans[2 * x + 2]; break;
	case '1': rans[x] = rans[2 * x + 1]; break;
	case '?': rans[x] = rans[2 * x + 2] + rans[2 * x + 1]; break;
	}
	return;
}
int32_t main() {
	IOS;
	int k;
	cin >> k;
	vector<char>val((1 << k)*2);
	rans.resize((1 << k)*2);
	string s;
	s.resize((1 << k));
	//bdt(g, 0, 0, (1 << k));
	for (int i=1; i < (1 << k); i++) {
		char c;
		cin >> c;
		int p = (1 << k) - i-1;
		val[p] = c;
		s[i] = c;
	}
	fcnt(0, val, 0, (1<<k));
	int q;
	cin >> q;
	while (q--) {
		int pos; char c;
		cin >> pos >> c;
		if (s[pos] != c) {
			int rpos = s.size() - pos-1;
			val[rpos] = c;
			ans(rpos, val);
		}
		cout << rans[0]<<"\n";
		s[pos] = c;
	}
}
### Prompt

Please provide a CPP coded solution to the problem described below:

2^k teams participate in a playoff tournament. The tournament consists of 2^k - 1 games. They are held as follows: first of all, the teams are split into pairs: team 1 plays against team 2, team 3 plays against team 4 (exactly in this order), and so on (so, 2^{k-1} games are played in that phase). When a team loses a game, it is eliminated, and each game results in elimination of one team (there are no ties). After that, only 2^{k-1} teams remain. If only one team remains, it is declared the champion; otherwise, 2^{k-2} games are played: in the first one of them, the winner of the game "1 vs 2" plays against the winner of the game "3 vs 4", then the winner of the game "5 vs 6" plays against the winner of the game "7 vs 8", and so on. This process repeats until only one team remains.

For example, this picture describes the chronological order of games with k = 3:

<image>

Let the string s consisting of 2^k - 1 characters describe the results of the games in chronological order as follows:

  * if s_i is 0, then the team with lower index wins the i-th game; 
  * if s_i is 1, then the team with greater index wins the i-th game; 
  * if s_i is ?, then the result of the i-th game is unknown (any team could win this game). 



Let f(s) be the number of possible winners of the tournament described by the string s. A team i is a possible winner of the tournament if it is possible to replace every ? with either 1 or 0 in such a way that team i is the champion.

You are given the initial state of the string s. You have to process q queries of the following form: 

  * p c — replace s_p with character c, and print f(s) as the result of the query. 

Input

The first line contains one integer k (1 ≤ k ≤ 18).

The second line contains a string consisting of 2^k - 1 characters — the initial state of the string s. Each character is either ?, 0, or 1.

The third line contains one integer q (1 ≤ q ≤ 2 ⋅ 10^5) — the number of queries.

Then q lines follow, the i-th line contains an integer p and a character c (1 ≤ p ≤ 2^k - 1; c is either ?, 0, or 1), describing the i-th query.

Output

For each query, print one integer — f(s).

Example

Input


3
0110?11
6
5 1
6 ?
7 ?
1 ?
5 ?
1 1


Output


1
2
3
3
5
4

### Response

```cpp
#pragma comment(linker, "/STACK: 268435456")
#pragma GCC optimization ("O3")
#pragma GCC optimization ("unroll-loops")
#pragma GCC optimize("Ofast")
#pragma GCC target("avx,avx2,fma")
#define _CRT_SECURE_NO_WARNINGS
#include <bits/stdc++.h>
#include <unordered_map>
#include <unordered_set>
#include <chrono>
using namespace std;
using namespace chrono;
#define IOS ios::sync_with_stdio(0); cin.tie(0); cout.tie(0);
#define int long long
#define double long double
#define sqr(x) (x)*(x)
#define pb push_back
typedef int C;
typedef complex<C>P;
#define X real()
#define Y imag()
#define ins insert
#define amas(a) for (auto&c:a) cout<<c<<" ";
#define pf push_front
#define pii pair<int,int>
struct custom_hash {
	static uint64_t splitmix64(uint64_t x) {
		x += 0x9e3779b97f4a7c15;
		x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
		x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
		return x ^ (x >> 31);
	}
	size_t operator()(uint64_t x) const {
		static const uint64_t fixed_random = steady_clock::now().time_since_epoch().count();
		return splitmix64(x + fixed_random);
	}
};
//void bdt(vector<vector<int>>& g, int x, int lx, int rx) { //[l;r)
//	if (rx - lx == 1) {
//		return;
//	}
//	int m = (lx + rx) / 2;
//	g[x].push_back(2 * x + 1);
//	g[2 * x + 1].push_back(x);
//	bdt(g, 2 * x + 1, lx, m);
//	g[x].push_back(2 * x + 2);
//	g[2 * x + 2].push_back(x);
//	bdt(g, 2 * x + 2, m, rx);
//}
vector<int>rans;
void ans(int x, vector<char>& val) {
	switch (val[x])
	{ 
	case '0': rans[x] = rans[2 * x + 2]; break;
	case '1': rans[x] = rans[2 * x + 1]; break;
	case '?': rans[x] = rans[2 * x + 2] + rans[2 * x + 1]; break;
	}
	if (x == 0) return;
	ans((x - 1) / 2, val);
}
void fcnt(int x, vector<char>& val,int lx,int rx) {
	if (rx - lx == 1) {
		rans[x] = 1;
		return;
	}
	int m = (lx + rx) / 2;
	fcnt(2 * x + 1, val, lx, m);
	fcnt(2 * x + 2, val, m, rx);
	switch (val[x])
	{
	case '0': rans[x] = rans[2 * x + 2]; break;
	case '1': rans[x] = rans[2 * x + 1]; break;
	case '?': rans[x] = rans[2 * x + 2] + rans[2 * x + 1]; break;
	}
	return;
}
int32_t main() {
	IOS;
	int k;
	cin >> k;
	vector<char>val((1 << k)*2);
	rans.resize((1 << k)*2);
	string s;
	s.resize((1 << k));
	//bdt(g, 0, 0, (1 << k));
	for (int i=1; i < (1 << k); i++) {
		char c;
		cin >> c;
		int p = (1 << k) - i-1;
		val[p] = c;
		s[i] = c;
	}
	fcnt(0, val, 0, (1<<k));
	int q;
	cin >> q;
	while (q--) {
		int pos; char c;
		cin >> pos >> c;
		if (s[pos] != c) {
			int rpos = s.size() - pos-1;
			val[rpos] = c;
			ans(rpos, val);
		}
		cout << rans[0]<<"\n";
		s[pos] = c;
	}
}
```