
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  $\mathbb{F}_p$  be the set of integers modulo  $p$ . Call a function  $f : \mathbb{F}_p^2 \to \mathbb{F}_p$  *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 the number of functions  $\mathbb{F}_p^2 \to \mathbb{F}_p$  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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