
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  $\alpha\not\equiv 0\pmod{p}$  such that
\[f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}\]
for all  $x, y\in\mathbb{Z}$ . How many good functions are there that are periodic with minimal period  $2016$ ?

*Ashwin Sah*

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