
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 be either alive or dead. Every minute, all live cells die simultaneously, and those dead cells that have an odd number of live neighbor cells (along the sides) come to life. Identify all pairs \((m, n)\) for which there exists an initial configuration of alive and dead cells such that life in the rectangle will persist forever (i.e., at every moment in time, at least one cell will be alive).

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