
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
Find all  $(m,n) \in \mathbb{Z}^2$  that we can color each unit square of  $m \times n$  with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.

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