Problem Statement

AtCoder Street is a road represented by a straight line on flat ground. There are $N$ utility poles of height $H$ standing on this road. The poles are numbered $1, 2, \dots, N$ in chronological order. Pole $i$ ($1 \leq i \leq N$) is vertically positioned at coordinate $X_i$. The base of each pole is fixed to the ground. Assume that the poles are sufficiently thin.

The street will experience $N$ earthquakes. During the $i$-th earthquake $(1 \leq i \leq N)$, the following events occur:

1. If pole $i$ has not yet fallen, it falls to the left or the right on the number line, each with a probability of $\frac{1}{2}$.
2. If a falling pole collides with another pole that has not yet fallen (including collision at the base of the pole), the latter pole will also fall in the same direction. This may trigger a chain reaction.

The direction in which a pole falls during step 1 is independent of the direction in which other poles have fallen.

The following figure is an example of how poles might fall during one earthquake:

For earthquake preparedness, for each $t = 1, 2, \dots, N$, find the probability that all poles have fallen by exactly the $t$-th earthquake. Multiply it by $2^N$ and print the result modulo $998244353$. It can be proved that the values to be printed are integers.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq H \leq 10^9$
• $0 \leq X_i \leq 10^9 \ (1 \leq i \leq N)$
• $X_1, X_2, \dots, X_N$ are all distinct.
• All input values are integers.

Input

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

N H
X_1 X_2 \cdots X_N

Output

Print the answers for $t = 1, 2, \dots, N$ in this order, separated by spaces.