
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  $N$  be the number of functions  $f: \mathbb{Z}/16\mathbb{Z} \to \mathbb{Z}/16\mathbb{Z}$  such that for all  $a,b \in \mathbb{Z}/16\mathbb{Z}$ : \[f(a)^2+f(b)^2+f(a+b)^2 \equiv 1+2f(a)f(b)f(a+b) \pmod{16}.\] Find the remainder when  $N$  is divided by 2017.

*Proposed by Zack Chroman*

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