The chess board industry has fallen on hard times and needs your help. It is a little-known fact that chess boards are made from the bark of the extremely rare Croatian Chess Board tree, (Biggus Mobydiccus). The bark of that tree is stripped and unwrapped into a huge rectangular sheet of chess board material. The rectangle is a grid of black and white squares.
Your task is to make as many large square chess boards as possible. A chess board is a piece of the bark that is a square, with sides parallel to the sides of the bark rectangle, with cells colored in the pattern of a chess board (no two cells of the same color can share an edge).
Each time you cut out a chess board, you must choose the largest possible chess board left in the sheet. If there are several such boards, pick the topmost one. If there is still a tie, pick the leftmost one. Continue cutting out chess boards until there is no bark left. You may need to go as far as cutting out 1-by-1 mini chess boards.
Here is an example showing the bark of a Chess Board tree and the first few chess boards that will be cut out of it.
The first line of the input gives the number of test cases, T. T test cases follow. Each one starts with a line containing the dimensions of the bark grid, M and N. N will always be a multiple of 4. The next M lines will each contain an (N/4)-character hexadecimal integer, representing a row of the bark grid. The binary representation of these integers will give you a strings of N bits, one for each row. Zeros represent black squares; ones represent white squares of the grid. The rows are given in the input from top to bottom. In each row, the most-significant bit of the hexadecimal integer corresponds to the leftmost cell in that row.
For each test case, output one line containing "Case #x: K", where x is the case number (starting from 1) and K is the number of different chess board sizes that you can cut out by following the procedure described above. The next K lines should contain two integers each -- the size of the chess board (from largest to smallest) and the number of chess boards of that size that you can cut out.
1 ≤ T ≤ 100;
N will be divisible by 4;
Each hexadecimal integer will contain exactly N/4 characters.
Only the characters 0-9 and A-F will be used.
1 ≤ M ≤ 32;
1 ≤ N ≤ 32.
1 ≤ M ≤ 512;
1 ≤ N ≤ 512;
The input file will be at most 200kB in size.
The first example test case represents the image above.