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F. Cupboards Jumps

time limit per test

6 secondsmemory limit per test

512 megabytesinput

standard inputoutput

standard outputIn the house where Krosh used to live, he had $$$n$$$ cupboards standing in a line, the $$$i$$$-th cupboard had the height of $$$h_i$$$. Krosh moved recently, but he wasn't able to move the cupboards with him. Now he wants to buy $$$n$$$ new cupboards so that they look as similar to old ones as possible.

Krosh does not remember the exact heights of the cupboards, but for every three consecutive cupboards he remembers the height difference between the tallest and the shortest of them. In other words, if the cupboards' heights were $$$h_1, h_2, \ldots, h_n$$$, then Krosh remembers the values $$$w_i = \max(h_{i}, h_{i + 1}, h_{i + 2}) - \min(h_{i}, h_{i + 1}, h_{i + 2})$$$ for all $$$1 \leq i \leq n - 2$$$.

Krosh wants to buy such $$$n$$$ cupboards that all the values $$$w_i$$$ remain the same. Help him determine the required cupboards' heights, or determine that he remembers something incorrectly and there is no suitable sequence of heights.

Input

The first line contains two integers $$$n$$$ and $$$C$$$ ($$$3 \leq n \leq 10^6$$$, $$$0 \leq C \leq 10^{12}$$$) — the number of cupboards and the limit on possible $$$w_i$$$.

The second line contains $$$n - 2$$$ integers $$$w_1, w_2, \ldots, w_{n - 2}$$$ ($$$0 \leq w_i \leq C$$$) — the values defined in the statement.

Output

If there is no suitable sequence of $$$n$$$ cupboards, print "NO".

Otherwise print "YES" in the first line, then in the second line print $$$n$$$ integers $$$h'_1, h'_2, \ldots, h'_n$$$ ($$$0 \le h'_i \le 10^{18}$$$) — the heights of the cupboards to buy, from left to right.

We can show that if there is a solution, there is also a solution satisfying the constraints on heights.

If there are multiple answers, print any.

Examples

Input

7 20 4 8 12 16 20

Output

YES 4 8 8 16 20 4 0

Input

11 10 5 7 2 3 4 5 2 1 8

Output

YES 1 1 6 8 6 5 2 0 0 1 8

Input

6 9 1 6 9 0

Output

NO

Note

Consider the first example:

- $$$w_1 = \max(4, 8, 8) - \min(4, 8, 8) = 8 - 4 = 4$$$
- $$$w_2 = \max(8, 8, 16) - \min(8, 8, 16) = 16 - 8 = 8$$$
- $$$w_3 = \max(8, 16, 20) - \min(8, 16, 20) = 20 - 8 = 12$$$
- $$$w_4 = \max(16, 20, 4) - \min(16, 20, 4) = 20 - 4 = 16$$$
- $$$w_5 = \max(20, 4, 0) - \min(20, 4, 0) = 20 - 0 = 20$$$

There are other possible solutions, for example, the following: $$$0, 1, 4, 9, 16, 25, 36$$$.

Codeforces (c) Copyright 2010-2021 Mike Mirzayanov

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