
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 set of positive integers. Let  $f: \mathbb N \to \mathbb N$  be a function such that the following conditions hold:

(a) For any  $n\in \mathbb N$ , we have  $f(n) | n^{2016}$ . 
(b) For any  $a,b,c\in \mathbb N$  satisfying  $a^2+b^2=c^2$ , we have  $f(a)f(b)=f(c)$ . 

Over all possible functions  $f$ , determine the number of distinct values that can be achieved by  $f(2014)+f(2)-f(2016)$ .

*Proposed by Vincent Huang*

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