
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 \(\mathbb{N}\) denote the natural numbers. Compute the number of functions \(f: \mathbb{N} \rightarrow\{0,1, \ldots, 16\}\) such that \(f(x+17)=f(x)\) and \(f\left(x^{2}\right) \equiv f(x)^{2}+15 \pmod{17}\) for all integers \(x \geq 1\).

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