Problem Statement

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.

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.

Input

The input is given from Standard Input in the following format:

L K N
A_1 A_2 \ldots A_N

Output

Output the answer.