
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 how many ways can we fill the cells of a  $m\times n$  board with  $+1$  and  $-1$  such that the product of numbers on each line and on each column are all equal to  $-1$ ?


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