Problem Statement

There are $N$ servals on a bridge of length $L$.

The $i$-th serval is located at position $A_{i}$ from the left end of the bridge.

Here, $0 < A_{1} < A_{2} < \cdots < A_{N} < L$ holds.

For each $i = 1, 2, \dots, N$, answer the following question:

The servals will perform the following three actions in order:

• Action $1$: The $N - 1$ servals other than the $i$-th serval face left or right.
• Action $2$: The $i$-th serval faces left or right.
• Action $3$: All servals start moving simultaneously. All servals move at a constant speed of exactly $1$ unit distance per unit time. When a serval reaches the end of the bridge, it leaves the bridge. If two servals collide, they both reverse their direction and continue moving.

The $i$-th serval is smart and loves this bridge, so when choosing a direction in Action $2$, it will observe the directions of the other $N-1$ servals and choose the direction that allows it to stay on the bridge the longer during Action $3$. There are $2^{N-1}$ possible combinations of directions for the $N-1$ servals in Action $1$. Find the sum, modulo $998244353$, over all these combinations, of the durations the $i$-th serval can stay on the bridge. It can be proved that the output value is an integer.

Constraints

• $1 \leq N \leq 10^{5}$
• $0 < A_{1} < A_{2} < \cdots < A_{N} < L \leq 10^{9}$
• All input values are integers.

Input

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

N L
A_{1} A_{2} \cdots A_{N}

Output

Print $N$ lines. The $k$-th line should contain the answer for $i=k$.