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D - Robot Arms 2

Score : $400$ points

Problem Statement

You are given a sequence $A = (A_1, A_2, \dots, A_N)$ of length $N$ consisting of positive integers, and integers $x$ and $y$.
Determine whether it is possible to place $N+1$ points $p_1, p_2, \dots, p_N, p_{N+1}$ in the $xy$-coordinate plane to satisfy all of the following conditions. (It is allowed to place two or more points at the same coordinates.)

• $p_1 = (0, 0)$.
• $p_2 = (A_1, 0)$.
• $p_{N+1} = (x, y)$.
• The distance between the points $p_i$ and $p_{i+1}$ is $A_i$. ($1 \leq i \leq N$)
• The segments $p_i p_{i+1}$ and $p_{i+1} p_{i+2}$ form a $90$ degree angle. ($1 \leq i \leq N - 1$)

Constraints

• $2 \leq N \leq 10^3$
• $1 \leq A_i \leq 10$
• $|x|, |y| \leq 10^4$
• All values in the input are integers.

Input

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

$N$ $x$ $y$

$A_1$ $A_2$ $\dots$ $A_N$

Output

If it is possible to place $p_1, p_2, \dots, p_N, p_{N+1}$ to satisfy all of the conditions in the Problem Statement, print Yes; otherwise, print No.

3 -1 1
2 1 3

Yes

The figure below shows a placement where $p_1 = (0, 0)$, $p_2 = (2, 0)$, $p_3 = (2, 1)$, and $p_4 = (-1, 1)$. All conditions in the Problem Statement are satisfied.

5 2 0
2 2 2 2 2

Yes

Letting $p_1 = (0, 0)$, $p_2 = (2, 0)$, $p_3 = (2, 2)$, $p_4 = (0, 2)$, $p_5 = (0, 0)$, and $p_6 = (2, 0)$ satisfies all the conditions. Note that multiple points may be placed at the same coordinates.

4 5 5
1 2 3 4

No

3 2 7
2 7 4

No

10 8 -7
6 10 4 1 5 9 8 6 5 1

Yes