
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
Let \( A = \left( a_{ij} \right) \) be the \( n \times n \) matrix, where \( a_{ij} \) is the remainder of the division of \( i^j + j^i \) by 3 for \( i, j = 1, 2, \ldots, n \). Find the greatest \( n \) for which \(\operatorname{det} A \ne 0\).

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