
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 an $m \times n$ rectangular grid filled with positive integers, an operation allows you to add an integer $k$ to both numbers in any adjacent cells, such that the resulting numbers are non-negative integers. Determine the necessary and sufficient condition for this operation to be able to make all numbers in the grid zero using a finite number of steps.
(30th IMO shortlist problem)

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