1461：Rotations and Reflections

Many games, tricks and puzzles depend on determining whether two patterns on a rectangular grid are the ``same'' or not. For instance, the 96 different ways of arranging 8 queens safely on a chessboard can be shown to consist of rotations and/or reflections of only 12 basic patterns.

Write a program that will read in pairs of patterns and determine whether there is a simple transformation that will convert one into the other. Because symmetrical patterns bear many relationships to each other, the transformations must be checked in a specific order. The possible transformations (in order) are:

Preservation:           The patterns are identical

90 degree rotation :    The pattern was rotated clockwise by 90 degrees

180 degree rotation:    The pattern was rotated clockwise by 180 degrees

270 degree rotation:    The pattern was rotated clockwise by 270 degrees

Reflection:             The pattern was reflected about the horizontal axis (effectively by a mirror held at the top of the pattern)

Combination:            A reflection (as above), followed by one of the above rotations

Improper:               The patterns do not match under any of the above transformations

Input will consist of a series of pairs of patterns. Each set will consist of a line containing a single integer N (2 <= N <= 10) giving the size of the patterns, followed by N lines. Each line will consist of N dots or `x's (specifying a line of the original pattern), a space, and another set of N dots and `x's (specifying a line of the transformed pattern). The file will be terminated by a line consisting of a single zero (0).

Output will consist of a series of lines, one for each pattern pair in the input. Each line will consist of one of the following: `Preserved', `Rotated through m degrees' (where m is one of 90, 180 or 270), `Reflected', `Reflected and rotated through m degrees', `Improper'.