
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 \( p \) be a prime and let \( \mathbb{F}_{p} \) be the set of integers modulo \( p \). Call a function \( f: \mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p} \) quasiperiodic if there exist \( a, b \in \mathbb{F}_{p} \), not both zero, so that \( f(x+a, y+b)=f(x, y) \) for all \( x, y \in \mathbb{F}_{p} \). Find, with proof, the number of functions \( \mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p} \) that can be written as the sum of some number of quasiperiodic functions.

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