
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
 $n\geq 2$  is a given integer. For two permuations  $(\alpha_1,\cdots,\alpha_n)$  and  $(\beta_1,\cdots,\beta_n)$  of  $1,\cdots,n$ , consider  $n\times n$  matrix  $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$  defined by  $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$ . Find every possible value of  $\det(A)$ .

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