Problem Statement

Given are positive integers N, K, and a sequence of K integers (A_1, A_2, \ldots, A_K).

Takahashi and Aoki will play a game with stones. Initially, there are some heaps of stones, each of which contains one or more stones. The players take turns doing the following operation, with Takahashi going first.

• Choose a heap with one or more stones remaining. Remove any number of stones between 1 and X (inclusive) from that heap, where X is the number of stones remaining.

The player who first gets unable to do the operation loses.

Now, consider the initial arrangements of stones satisfying the following.

• 1\leq M\leq N holds, where M is the number of heaps of stones.
• The number of stones in each heap is one of the following: A_1, A_2, \ldots, A_K.

Assuming that the heaps are ordered, there are K+K^2+\cdots +K^N such initial arrangements of stones. Among them, find the number, modulo 998244353, of arrangements that lead to Takahashi's win, assuming that both players play optimally.

Constraints

• 1 \leq N \leq 2\times 10^5
• 1 \leq K < 2^{16}
• 1 \leq A_i < 2^{16}
• All A_i are distinct.
• All values in input are integers.

Sample Input 1

2 2
1 2

Sample Output 1

4

There are six possible initial arrangements of stones: (1), (2), (1,1), (1,2), (2,1), and (2,2).
Takahashi has a winning strategy for four of them: (1), (2), (1,2), and (2,1), and Aoki has a winning strategy for the other two. Thus, we should print 4.

Sample Input 2

100 3
3 5 7

Sample Output 2

112184936

Be sure to find the count modulo 998244353.