
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  $p = 2^{16}+1$  be a prime, and let  $S$  be the set of positive integers not divisible by  $p$ .
Let  $f: S \to \{0, 1, 2, ..., p-1\}$  be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all  $x,y \in S$ .
Let  $N$  be the product of all possible nonzero values of  $f(81)$ .
Find the remainder when when  $N$  is divided by  $p$ .

*Proposed by Yang Liu and Ryan Alweiss*

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