
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  $p = 2017$  be a prime. Given a positive integer  $n$ , let  $T$  be the set of all  $n\times n$  matrices with entries in  $\mathbb{Z}/p\mathbb{Z}$ . A function  $f:T\rightarrow \mathbb{Z}/p\mathbb{Z}$  is called an  $n$ -*determinant* if for every pair  $1\le i, j\le n$  with  $i\not= j$ , \[f(A) = f(A'),\] where  $A'$  is the matrix obtained by adding the  $j$ th row to the  $i$ th row.

Let  $a_n$  be the number of  $n$ -determinants. Over all  $n\ge 1$ , how many distinct remainders of  $a_n$  are possible when divided by  $\dfrac{(p^p - 1)(p^{p - 1} - 1)}{p - 1}$ ?

*Proposed by Ashwin Sah*

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