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F. A Random Code Problem

time limit per test

3 secondsmemory limit per test

512 megabytesinput

standard inputoutput

standard outputYou are given an integer array $$$a_0, a_1, \dots, a_{n - 1}$$$, and an integer $$$k$$$. You perform the following code with it:

`long long ans = 0; // create a 64-bit signed variable which is initially equal to 0`

for(int i = 1; i <= k; i++)

{

int idx = rnd.next(0, n - 1); // generate a random integer between 0 and n - 1, both inclusive

// each integer from 0 to n - 1 has the same probability of being chosen

ans += a[idx];

a[idx] -= (a[idx] % i);

}

Your task is to calculate the expected value of the variable ans after performing this code.

Note that the input is generated according to special rules (see the input format section).

Input

The only line contains six integers $$$n$$$, $$$a_0$$$, $$$x$$$, $$$y$$$, $$$k$$$ and $$$M$$$ ($$$1 \le n \le 10^7$$$; $$$1 \le a_0, x, y < M \le 998244353$$$; $$$1 \le k \le 17$$$).

The array $$$a$$$ in the input is constructed as follows:

- $$$a_0$$$ is given in the input;
- for every $$$i$$$ from $$$1$$$ to $$$n - 1$$$, the value of $$$a_i$$$ can be calculated as $$$a_i = (a_{i - 1} \cdot x + y) \bmod M$$$.

Output

Let the expected value of the variable ans after performing the code be $$$E$$$. It can be shown that $$$E \cdot n^k$$$ is an integer. You have to output this integer modulo $$$998244353$$$.

Examples

Input

3 10 3 5 13 88

Output

382842030

Input

2 15363 270880 34698 17 2357023

Output

319392398

Note

The array in the first example test is $$$[10, 35, 22]$$$. In the second example, it is $$$[15363, 1418543]$$$.

Codeforces (c) Copyright 2010-2022 Mike Mirzayanov

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