
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
In a grid rectangle of size \( m \times n \), each cell can either be alive or dead. Every minute, all living cells simultaneously die, while the dead cells that have an odd number of alive neighbors (by side) come to life.

List all pairs \((m, n)\) for which there exists an initial configuration of alive and dead cells such that life in the rectangle will exist forever (i.e., at least one cell will be alive at any given time)?

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