
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  $p > 2$  be a prime number.  $\mathbb{F}_p[x]$  is defined as the set of polynomials in  $x$  with coefficients in  $\mathbb{F}_p$  (the integers modulo  $p$  with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of  $x^k$  are equal in  $\mathbb{F}_p$  for each nonnegative integer  $k$ . For example,  $(x+2)(2x+3) = 2x^2 + 2x + 1$  in  $\mathbb{F}_5[x]$  because the corresponding coefficients are equal modulo 5.

Let  $f, g \in \mathbb{F}_p[x]$ . The pair  $(f, g)$  is called *compositional* if
\[f(g(x)) \equiv x^{p^2} - x\]
in  $\mathbb{F}_p[x]$ . Find, with proof, the number of compositional pairs.

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