
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
The points on an $m \times n$ grid can be either white or black. A grid is said to be balanced if for any point $P$ on it, the row and column that pass through this point $P$ both have the same number of points of the same color as $P$. Determine all pairs of positive integers $(m, n)$ for which there exists a balanced grid.

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