
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  $A=(a_{ij})$  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  $\det A\ne0$ .

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