AT_agc073_a [AGC073A] Chords and Checkered

You are given integers $ L, K $ and a length- $ N $ non-negative integer sequence $ A = (A_1, A_2, \ldots, A_N) $ . Here, $ 2K < L $ and $ 0 \leq A_1 < A_2 < \cdots < A_N < L $ are guaranteed. Consider a circle with circumference $ L $ . Take an arbitrary point on this circle as the reference point, and call the point reached by traveling clockwise distance $ x $ along the circumference from there the point with coordinate $ x $ . Also, for each $ i $ ( $ 1 \leq i \leq N $ ), consider the chord connecting the point with coordinate $ A_i $ and the point with coordinate $ A_i + K $ , and call this chord $ i $ . For a set of chords $ s $ , the **score** is determined by the following procedure: - Draw all chords contained in $ s $ . The drawn chords divide the circle into several regions; color them with white and black. First, color the region containing the center of the circle white, and color the remaining regions so that adjacent regions (connected by a line segment of positive length) have different colors. It can be proved that such a coloring method always exists uniquely. The score is the number of regions colored black. There are $ 2^N $ possible sets for $ s $ . Find the sum, modulo $ 998244353 $ , of scores for all of these.

The input is given from Standard Input in the following format: > $ L $ $ K $ $ N $ $ A_1 $ $ A_2 $ $ \ldots $ $ A_N $

Output the answer.

### Sample Explanation 1 - When no chords are chosen, the score is $ 0 $ . - When chord $ 1 $ is chosen, the score is $ 1 $ . - When chord $ 2 $ is chosen, the score is $ 1 $ . - When chords $ 1, 2 $ are chosen, the score is $ 2 $ . Therefore, the answer is $ 4 $ , which is the sum of these. ### Constraints - $ 1 \leq K $ - $ 2K < L \leq 10^9 $ - $ 1 \leq N \leq 250000 $ - $ 0 \leq A_1 < A_2 < \cdots < A_N < L $ - All input values are integers.