
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 all 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 (in terms of \( p \)).

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