#include <bits/stdc++.h>
using namespace std;
long long mod(long long x) {
  if (x < 0) {
    x *= -1;
    x %= 1000000009;
    x *= -1;
    x += 1000000009;
  }
  return x % 1000000009;
}
long long fastexp(long long base, long long exp) {
  if (exp == 0) return 1;
  long long temp = fastexp(base, exp / 2);
  temp *= temp;
  temp = mod(temp);
  if (exp % 2 == 1) temp *= base;
  temp = mod(temp);
  return temp;
}
long long inverse(long long x) { return fastexp(x, 1000000009 - 2); }
int main() {
  long long n, a, b, k, ans = 0;
  string s;
  int i;
  cin >> n >> a >> b >> k;
  cin >> s;
  for (i = 0; i < s.length(); i++) {
    if (s[i] == '+') {
      ans += (fastexp(a, n - i) * fastexp(b, i));
      ans = mod(ans);
    } else {
      ans -= (fastexp(a, n - i) * fastexp(b, i));
      ans = mod(ans);
    }
  }
  long long x = fastexp(a, k);
  x = mod(x);
  long long y = fastexp(b, k);
  y = mod(y);
  long long total = 0;
  if (x == y) {
    total = ans * ((n + 1) / k);
    total = mod(total);
  } else {
    total = fastexp(mod(y * inverse(x)), (n + 1) / k);
    total = mod(total);
    total = ans * (total - 1);
    total = mod(total);
    total *= inverse((mod(y * inverse(x))) - 1);
    total = mod(total);
  }
  total = mod(total);
  cout << total << endl;
  return 0;
}

### Prompt

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

You are given two integers a and b. Moreover, you are given a sequence s_0, s_1, ..., s_{n}. All values in s are integers 1 or -1. It's known that sequence is k-periodic and k divides n+1. In other words, for each k ≤ i ≤ n it's satisfied that s_{i} = s_{i - k}.

Find out the non-negative remainder of division of ∑ _{i=0}^{n} s_{i} a^{n - i} b^{i} by 10^{9} + 9.

Note that the modulo is unusual!

Input

The first line contains four integers n, a, b and k (1 ≤ n ≤ 10^{9}, 1 ≤ a, b ≤ 10^{9}, 1 ≤ k ≤ 10^{5}).

The second line contains a sequence of length k consisting of characters '+' and '-'. 

If the i-th character (0-indexed) is '+', then s_{i} = 1, otherwise s_{i} = -1.

Note that only the first k members of the sequence are given, the rest can be obtained using the periodicity property.

Output

Output a single integer — value of given expression modulo 10^{9} + 9.

Examples

Input

2 2 3 3
+-+


Output

7


Input

4 1 5 1
-


Output

999999228

Note

In the first example:

(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = 2^{2} 3^{0} - 2^{1} 3^{1} + 2^{0} 3^{2} = 7

In the second example:

(∑ _{i=0}^{n} s_{i} a^{n - i} b^{i}) = -1^{4} 5^{0} - 1^{3} 5^{1} - 1^{2} 5^{2} - 1^{1} 5^{3} - 1^{0} 5^{4} = -781 ≡ 999999228 \pmod{10^{9} + 9}.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
long long mod(long long x) {
  if (x < 0) {
    x *= -1;
    x %= 1000000009;
    x *= -1;
    x += 1000000009;
  }
  return x % 1000000009;
}
long long fastexp(long long base, long long exp) {
  if (exp == 0) return 1;
  long long temp = fastexp(base, exp / 2);
  temp *= temp;
  temp = mod(temp);
  if (exp % 2 == 1) temp *= base;
  temp = mod(temp);
  return temp;
}
long long inverse(long long x) { return fastexp(x, 1000000009 - 2); }
int main() {
  long long n, a, b, k, ans = 0;
  string s;
  int i;
  cin >> n >> a >> b >> k;
  cin >> s;
  for (i = 0; i < s.length(); i++) {
    if (s[i] == '+') {
      ans += (fastexp(a, n - i) * fastexp(b, i));
      ans = mod(ans);
    } else {
      ans -= (fastexp(a, n - i) * fastexp(b, i));
      ans = mod(ans);
    }
  }
  long long x = fastexp(a, k);
  x = mod(x);
  long long y = fastexp(b, k);
  y = mod(y);
  long long total = 0;
  if (x == y) {
    total = ans * ((n + 1) / k);
    total = mod(total);
  } else {
    total = fastexp(mod(y * inverse(x)), (n + 1) / k);
    total = mod(total);
    total = ans * (total - 1);
    total = mod(total);
    total *= inverse((mod(y * inverse(x))) - 1);
    total = mod(total);
  }
  total = mod(total);
  cout << total << endl;
  return 0;
}

```