
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
We consider an  $n \times n$  table, with  $n\ge1$ . Aya wishes to color  $k$  cells of this table so that that there is a unique way to place  $n$  tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of  $k$  for which Aya's wish is achievable?

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