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## Learning Opportunities

This puzzle can be solved using the following concepts. Practice using these concepts and improve your skills.

# The Goal

It worked! Oh wait... Detective Pikaptcha is now trapped in a more advanced type of space-warp maze. It looks like a ring but twisted. It doesn’t take long for Pikaptcha to realize that the maze is actually a gigantic Möbius strip.

“Let’s try the same method...”

Your objective is to write a program that will compute, for each cell of a maze, the number of times Pikaptcha will step into the cell by following a wall until he reaches his original location.

# Topology

Pikaptcha looked up his database which responded, "A Möbius strip is a surface with only one side and only one edge." But wait...Pikaptcha found that this one is not the standard Möbius because the edge is gone. It is a variant having one endless looping surface and no edge. Pikaptcha can cross the edge to go to the other side of the strip if it is not blocked by walls.

The Möbius strip maze is given to you as a grid filled with 0s and #s, where 0 represents a passage, and # represents a wall: an impassable cell.

Here are instructions to create a real-world representation of a Möbius maze:

• Print the first half (along the x-axis) of the grid on a piece of paper.
• Flip the paper over vertically.
• Print the rest of the grid on the side now in front of you.
• Give the strip a half-twist and connect the two ends to create a loop.

# Rules

The initial position and direction of Pikaptcha is given to you in the grid as a special character:
• >: facing right
• v: facing down
• <: facing left
• ^: facing up
An additional character indicates which wall Pikaptcha must follow:
• R for the wall on his right
• L for the wall on his left

We’re considering the 4-adjacency, meaning a cell has a maximum of 4 adjacent cells (a diagonal cell is not adjacent).

Important
Pikaptcha can step over the edge of the Möbius strip. The two sides of the strip are in fact belonging to the same surface which was folded up. Pikaptcha will follow the same wall-following rules in his journey no matter he is on a flat surface or crossing the edge.
You must analyze the given maze and return it with a small transformation: for each empty cell, instead of a 0, you must return the number of times Pikaptcha stepped into that cell while striding along the maze, following a wall. For each impassable cell, you change nothing: you still return #.

# Game Protocol

Game Input
First line: 2 integers width and height for the size of the Möbius strip.
Next height lines: a string line of length width where 0 is a passage and # is a wall and >, v, < or ^ is the initial position of Pikaptcha.
Next line: a character side for which wall to follow (from Pikaptcha's perspective).
Game Output
height line of width characters each containing the transformed grid.
Constraints

1width & height200

Allotted response time to output is ≤ 2s

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