
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  $\mathbb{Z}_{\geq 0}$  be the set of nonnegative integers. Let  $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$  be a function such that, for all  $a,b \in \mathbb{Z}_{\geq0}$ : \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\] 
Furthermore, suppose there exists  $n \in \mathbb{Z}_{\geq0}$  such that  $f(n)=577$ . Let  $S$  be the sum of all possible values of  $f(2017)$ . Find the remainder when  $S$  is divided by  $2017$ .

*Proposed by Zack Chroman*

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