
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 \( p=2017 \) be a prime and \( \mathbb{F}_{p} \) be the integers modulo \( p \). A function \( f: \mathbb{Z} \rightarrow \mathbb{F}_{p} \) is called good if there is \( \alpha \in \mathbb{F}_{p} \) with \( \alpha \not \equiv 0(\bmod p) \) such that
$$
f(x) f(y)=f(x+y)+\alpha^{y} f(x-y) \quad(\bmod p)
$$
for all \( x, y \in \mathbb{Z} \). How many good functions are there that are periodic with minimal period 2016?

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