
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 call a monic polynomial  $P(x) \in \mathbb{Z}[x]$  *square-free mod n* if there <u>dose not</u> exist polynomials  $Q(x),R(x) \in \mathbb{Z}[x]$  with  $Q$  being non-constant and  $P(x) \equiv Q(x)^2 R(x) \mod n$ . Given a prime  $p$  and integer  $m \geq 2$ . Find the number of monic *square-free mod p*  $P(x)$  with degree  $m$  and coeeficients in  $\{0,1,2,3,...,p-1\}$ .

*Proposed by Masud Shafaie*

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