Problem Statement

There is an infinitely large two-dimensional grid. The color of cell $(x,y)$ is black if $0 \leq x \leq X-1$ and $0 \leq y \leq Y-1$, and white otherwise.

$N$ great storms occur in order on this grid. The $i$-th great storm updates the color of each cell on the grid according to a rule represented by a character $C_i$ and integers $A_i, B_i$.

In the $i$-th great storm, the color of cell $(x, y)$ after the great storm is:

• In case $C_i$ is X,
• if $x < A_i$, it becomes the color of cell $(x, y + B_i)$ before the great storm;
  • if $x \geq A_i$, it becomes the color of cell $(x, y - B_i)$ before the great storm.
• In case $C_i$ is Y,
• if $y < A_i$, it becomes the color of cell $(x + B_i, y)$ before the great storm;
  • if $y \geq A_i$, it becomes the color of cell $(x - B_i, y)$ before the great storm.

Two black cells $(x_1, y_1), (x_2, y_2)$ are adjacent if and only if $|x_1 - x_2| + |y_1 - y_2| = 1$. Also, two black cells $c_1, c_2$ are connected if and only if one can move from cell $c_1$ to cell $c_2$ by repeatedly moving to adjacent black cells.

A non-empty set $S$ of black cells is a connected component if and only if $S$ satisfies the following conditions:

• Any two cells in $S$ are connected.
• Every black cell not in $S$ is not connected to any cell in $S$.

For the grid after all $N$ great storms have occurred, find the number of connected components and output the number of cells in each connected component in ascending order.

Constraints

• $N, X, Y$ are integers.
• $1 \leq N \leq 14$
• $1 \leq X,Y \leq 10^8$
• $C_i$ is X or Y.
• $A_i, B_i$ are integers.
• $-10^8 \leq A_i, B_i \leq 10^8$

Input

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

N X Y
C_1 A_1 B_1
\vdots
C_N A_N B_N

Output

Output two lines.

The first line should contain the number of connected components consisting of black cells.

The second line should contain the number of cells in each connected component, space-separated, in ascending order.