Put a ball in the bottom left corner of a rectangular billiards table,
with pockets (holes) in each of the four corners. Then cue the ball in a
45° angle away from the corner pocket. Assume that there is no
friction, so the ball never slows down, and also assume that the ball
perfectly bounces off at the cushions (in a 90° angle). Where does
the ball touch the cushions and in what corner pocket does it disappear?

A billiards table with height 6 and width 8, with named cushions and pockets.

To describe the positions of the ball, we assume that the billiards table
has integer dimensions: the height $$h$$ and the width $$w$$ of the table
are natural numbers (the units are not important for this assignment). As
indicated in the figure above, we overlay an imaginary coordinate system
on top of the billiards table, with the X-axis along the bottom cushion
and the Y-axis along the left cushion. This way, we can describe the
positions where the ball hits the cushions using $$(x, y)$$-coordinates,
where $$x, y \in \mathbb{N}$$ and $$0 \leq x \leq w, 0 \leq y \leq h$$.
The origin $$(0, 0)$$ of the coordinate system is in the bottom left
pocket of the billiards table.

Input

Two lines that respectively contain the height $$h \in \mathbb{N}_0$$ and
the width $$w \in \mathbb{N}_0$$ of the billiards table.

Output

Generate a line for each successive cushion hit by the ball, containing
the name of the cushion, followed by a space and the $$(x, y)$$ coordinate
where the ball hits the cushion. Cushions are named as indicated in the
figure above.

Generate a last line containing the name of the pocket where the ball
disappears from the table, followed by a space and the $$(x, y)$$
coordinate of that pocket. Pockets are named as indicated in the figure
above.

Example

Input:

6
8

Output:

top cushion (6, 6)
right cushion (8, 4)
bottom cushion (4, 0)
left cushion (0, 4)
top cushion (2, 6)
bottom right pocket (8, 0)

Epilogue

In 2011 Royal
Caribbean^{1} introduced a self-leveling pool table on its
cruise ship Radiance of the Seas^{2}. This video was shot
during a storm in the Pacific — gyroscopes inside the table keep
the playing surface level so a game can continue even as the ship rolls.