AT_fps_24_j スゴロク

There is a board game with $ N+1 $ squares numbered $ 0, 1, \dots, N $ . You also have an $ M $ -sided die, where each side shows a distinct positive integer $ A_1, A_2, \dots, A_M $ with equal probability. Additionally, there are traps on squares $ B_1, B_2, \dots, B_L $ . Here, $ 1 \leq B_i \leq N-1 $ , and all $ B_i $ are distinct. You will play the following game: - Place your piece on square $ 0 $ at the beginning. While the game continues, repeat: - Roll the die. If your piece is on square $ x $ and the die shows $ y $ , move the piece to square $ \min(N, x+y) $ . - If the piece lands on a trap square, you lose immediately. - If the piece reaches square $ N $ without landing on a trap, you win immediately. Compute the probability that you win the game, modulo $ 998244353 $ . What does probability modulo $ 998244353 $ mean? It can be proven that the probability is always a rational number. Under the constraints of this problem, if the probability is written as $ \tfrac{P}{Q} $ with coprime integers $ P $ and $ Q $ , then there exists a unique integer $ R $ such that $ R \times Q \equiv P \pmod{998244353} $ and $ 0 \leq R < 998244353 $ . You are asked to compute this $ R $ .

The input is given from standard input in the following format: > $ N $ $ M $ $ L $ $ A_1 $ $ A_2 $ $ \dots $ $ A_M $ $ B_1 $ $ B_2 $ $ \dots $ $ B_L $

Print the probability that you win the game, modulo $ 998244353 $ .

### Sample Explanation 1 You can only win if the die shows $ 2 $ . Thus, the probability is $ \tfrac{1}{2} $ . ### Constraints - $ 2 \leq N \leq 2.5 \times 10^5 $ - $ 1 \leq M \leq N $ - $ 1 \leq L \leq N-1 $ - $ 1 \leq A_1 < A_2 < \dots < A_M \leq N $ - $ 1 \leq B_1 < B_2 < \dots < B_L \leq N-1 $ - All input values are integers