
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
Find all integer  $c\in\{0,1,...,2016\}$  such that the number of  $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$  which satisfy the following condition is minimal:
(1)  $f$  has periodic  $2017$ (2)  $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$ Proposed by William Chao

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