D - Robot Arms

Score : $600$ points

Problem Statement

Snuke is introducing a robot arm with the following properties to his factory:

• The robot arm consists of $m$ sections and $m+1$ joints. The sections are numbered $1$, $2$, ..., $m$, and the joints are numbered $0$, $1$, ..., $m$. Section $i$ connects Joint $i-1$ and Joint $i$. The length of Section $i$ is $d_i$.
• For each section, its mode can be specified individually. There are four modes: L, R, D and U. The mode of a section decides the direction of that section. If we consider the factory as a coordinate plane, the position of Joint $i$ will be determined as follows (we denote its coordinates as $(x_i, y_i)$):
  • $(x_0, y_0) = (0, 0)$.
  • If the mode of Section $i$ is L, $(x_{i}, y_{i}) = (x_{i-1} - d_{i}, y_{i-1})$.
  • If the mode of Section $i$ is R, $(x_{i}, y_{i}) = (x_{i-1} + d_{i}, y_{i-1})$.
  • If the mode of Section $i$ is D, $(x_{i}, y_{i}) = (x_{i-1}, y_{i-1} - d_{i})$.
  • If the mode of Section $i$ is U, $(x_{i}, y_{i}) = (x_{i-1}, y_{i-1} + d_{i})$.

Snuke would like to introduce a robot arm so that the position of Joint $m$ can be matched with all of the $N$ points $(X_1, Y_1), (X_2, Y_2), ..., (X_N, Y_N)$ by properly specifying the modes of the sections. Is this possible? If so, find such a robot arm and how to bring Joint $m$ to each point $(X_j, Y_j)$.

Constraints

• All values in input are integers.
• $1 \leq N \leq 1000$
• $-10^9 \leq X_i \leq 10^9$
• $-10^9 \leq Y_i \leq 10^9$

Partial Score

• In the test cases worth $300$ points, $-10 \leq X_i \leq 10$ and $-10 \leq Y_i \leq 10$ hold.

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Input

Input is given from Standard Input in the following format:

$N$

$X_1$ $Y_1$

$X_2$ $Y_2$

$:$

$X_N$ $Y_N$

Output

If the condition can be satisfied, follow the following format. If the condition cannot be satisfied, print -1.

``` $m$ $d_1$ $d_2$ $...$ $d_m$ $w_1$ $w_2$ $:$ $w_N$ ```

$m$ and $d_i$ are the configurations of the robot arm. Refer to the problem statement for what each of them means. Here, $1 \leq m \leq 40$ and $1 \leq d_i \leq 10^{12}$ must hold. Also, $m$ and $d_i$ must all be integers.

$w_j$ is a string of length $m$ that represents the way to bring Joint $m$ of the robot arm to point $(X_j, Y_j)$. The $i$-th character of $w_j$ should be one of the letters L, R, D and U, representing the mode of Section $i$.

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3
-1 0
0 3
2 -1

2
1 2
RL
UU
DR

In the given way to bring Joint $m$ of the robot arm to each $(X_j, Y_j)$, the positions of the joints will be as follows:

• To $(X_1, Y_1) = (-1, 0)$: First, the position of Joint $0$ is $(x_0, y_0) = (0, 0)$. As the mode of Section $1$ is R, the position of Joint $1$ is $(x_1, y_1) = (1, 0)$. Then, as the mode of Section $2$ is L, the position of Joint $2$ is $(x_2, y_2) = (-1, 0)$.
• To $(X_2, Y_2) = (0, 3)$: $(x_0, y_0) = (0, 0), (x_1, y_1) = (0, 1), (x_2, y_2) = (0, 3)$.
• To $(X_3, Y_3) = (2, -1)$: $(x_0, y_0) = (0, 0), (x_1, y_1) = (0, -1), (x_2, y_2) = (2, -1)$.

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5
0 0
1 0
2 0
3 0
4 0

-1

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2
1 1
1 1

2
1 1
RU
UR

There may be duplicated points among $(X_j, Y_j)$.

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3
-7 -3
7 3
-3 -7

5
3 1 4 1 5
LRDUL
RDULR
DULRD