Kitty's Light Switching Challenge

Kitty has $n^2$ lights arranged in an $n \times n$ matrix. Some of these lights are on and some are off. Kitty can perform three types of operations:

1. Choose a row and reverse the state of every light in that row.
2. Choose a column and reverse the state of every light in that column.
3. Choose exactly one light and reverse its state; this operation can be used at most $k$ times.

The goal is to turn off all the lights using at most $3n$ operations. The operations should be output in the following format (using 1-indexed positions):

• Row reversal: R i (toggles row $i$)
• Column reversal: C j (toggles column $j$)
• Single light toggle: L i j (toggles the light at row $i$, column $j$)

If a valid sequence exists that satisfies the constraints (i.e. the number of single light toggles does not exceed $k$, and the total number of operations is at most $3n$), output the number of operations used followed by the list of operations. Otherwise, output -1.

inputFormat

The first line contains two integers $n$ and $k$, denoting the size of the matrix and the maximum number of individual light toggles allowed.\nEach of the following $n$ lines contains $n$ integers (each 0 or 1), representing the state of the lights (0 for off, 1 for on).

outputFormat

If a valid sequence of operations exists, first output an integer $m$, the total number of operations performed. Then output $m$ lines, each describing one operation in the order they are performed. If no valid sequence exists, output -1.

sample

2 2
0 1
1 0

2
R 2
C 2