E. Special Positionstime limit per test5 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given an array a of length n. Also you are given m distinct positions p_1, p_2, \ldots, p_m (1 \leq p_i \leq n).A non-empty subset of these positions T is randomly selected with equal probability and the following value is calculated: \sum_{i=1}^{n} (a_i \cdot \min_{j \in T} \left|i - j\right|). In other word, for each index of the array, a_i and the distance to the closest chosen position are multiplied, and then these values are summed up.Find the expected value of this sum.This value must be found modulo 998\,244\,353. More formally, let M = 998\,244\,353. It can be shown that the answer can be represented as an irreducible fraction \frac{p}{q}, where p and q are integers and q \neq 0 (mod M). Output the integer equal to p \cdot q^{-1} (mod M). In other words, output such integer x that 0 \leq x < M and x \cdot q = p (mod M).InputThe first line contains two integers n and m (1 \leq m \leq n \leq 10^5).The second line contains n integers a_1, a_2, \ldots, a_n (0 \leq a_i < 998\,244\,353).The third line contains m distinct integers p_1, p_2, \ldots, p_m (1 \leq p_i \le n).For every 1 \leq i < m it is guaranteed that p_i < p_{i+1}.OutputPrint a single integer — the answer to the problem.ExamplesInput
4 2
1 2 3 4
1 4
Output
665496247
Input
6 6
4 2 4 2 4 2
1 2 3 4 5 6
Output
855638030
NoteIn the first test:   If only 1 is choosen, than the value equals to 1 \cdot 0 + 2 \cdot 1 + 3 \cdot 2 + 4 \cdot 3 = 20.  If only 4 is choosen, than the value equals to 1 \cdot 3 + 2 \cdot 2 + 3 \cdot 1 + 4 \cdot 0 = 10.  If both positions are chosen, than the value equals to 1 \cdot 0 + 2 \cdot 1 + 3 \cdot 1 + 4 \cdot 0 = 5. The answer to the problem is \frac{20 + 10 + 5}{3} = \frac{35}{3} = 665\,496\,247 (modulo 998\,244\,353).
4 2
1 2 3 4
1 4

665496247 
['combinatorics', 'divide and conquer', 'fft', 'math', '*3300']