Score : $525$ points

问题描述

一个机器人位于坐标平面的原点，其中正 $x$ 轴指向右侧，正 $y$ 轴指向上方。最初，机器人面向正 $x$ 轴方向。

您将按照以下顺序执行从 $i=1,\ldots,N$ 的操作：

• 使机器人顺时针或逆时针旋转 $90$ 度。然后，机器人按照它所面对的方向向前移动 $A_i$ 个单位。

是否存在一种选择旋转方向的方式，使得在 $N$ 次操作后，机器人能够到达坐标 $(X,Y)$？

如果可能，请确定每次操作应选择顺时针还是逆时针方向。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

A robot is at the origin of a coordinate plane where the positive $x$-axis points to the right and the positive $y$-axis points upwards. Initially, the robot is facing the positive $x$-direction.

You will perform the following operation for $i=1,\ldots,N$ in this order.

• Rotate the robot $90$ degrees clockwise or counterclockwise. Then, the robot moves forward $A_i$ units in the direction it is facing.

Can the direction of rotation be chosen so that the robot will be at the coordinates $(X,Y)$ after the $N$ operations?

If it is possible, determine which direction, clockwise or counterclockwise, should be chosen for each operation.

Constraints

• $1 \leq N \leq 80$
• $1 \leq A_i \leq 10^7$
• $-10^9\leq X,Y \leq 10^9$
• All input values are integers.

Input

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

$N$ $X$ $Y$

$A_1$ $\ldots$ $A_N$

Output

If the robot cannot be at the coordinates $(X,Y)$ after the $N$ operations, print No.
If the robot can be at the coordinates $(X,Y)$ after the $N$ operations, the first line should contain Yes, and the second line should contain a string $S$ of length $N$ that satisfies the following condition.
Condition: $S$ consists of L and R, and the following choices can put the robot at the coordinates $(X,Y)$ after the $N$ operations: in the $i$-th operation, rotate the robot counterclockwise if the $i$-th character of $S$ is L, and rotate it clockwise if it is R.

If there are multiple solutions, you may print any of them.

Sample Input 1

3 -2 4
3 2 1

Sample Output 1

Yes
LLR

Initially, the robot is at $(0,0)$ and facing the positive $x$-direction. The following choices can put the robot at the coordinates $(X,Y)$ after the $N$ operations.

• First operation: Rotate the robot $90$ degrees counterclockwise to face the positive $y$-direction. The robot moves forward $A_1=3$ units in the direction it is facing, and moves to $(0,3)$.
• Second operation: Rotate the robot $90$ degrees counterclockwise to face the negative $x$-direction. The robot moves forward $A_2=2$ units in the direction it is facing, and moves to $(-2,3)$.
• Third operation: Rotate the robot $90$ degrees clockwise to face the positive $y$-direction. The robot moves forward $A_3=1$ unit in the direction it is facing, and moves to $(-2,4)$.

Sample Input 2

1 0 0
1

Sample Output 2

No

Sample Input 3

4 0 0
1 1 1 1

Sample Output 3

Yes
LRRR

Outputs such as LLLL and RRRR are also accepted.

Sample Input 4

14 2543269 -1705099
3 14 159 2653 58979 323846 2643383 2795028 841971 69399 37510 58 20 9

Sample Output 4

Yes
LLLLLLLLLRLRRR

update @ 2024/3/10 01:50:08