F. A Random Code Problem
time limit per test
3 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

You 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 $$$[15363, 1418543]$$$.