
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
Some boys and girls form an \(n \times n\) grid. We know the number of girls in each row, column, and along each line parallel to the diagonals. For which values of \(n\) is this information sufficient to determine the exact positions of the girls in the grid? For which positions can we definitively state whether there is a girl there?

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