
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider a board of  $a \times b$ , with  $a$  and  $b$  integers greater than or equal to  $2$ . Initially their squares are colored black and white like a chess board. The permitted operation consists of choosing two squares with a common side and recoloring them as follows: a white square becomes black; a black box turns green; a green box turns white. Determine for which values of  $a$  and  $b$  it is possible, by a succession of allowed operations, to make all the squares that were initially white end black and all the squares that were initially black end white. 

Clarification: Initially there are no green squares, but they appear after the first operation.

Present the answer in LaTex format: \boxed{Your answer}
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