
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
Define a function \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) such that for any \( x, y \in \mathbf{Z} \), the following holds:
\[ f\left(x^{2}-3 y^{2}\right) + f\left(x^{2}+y^{2}\right) = 2(x+y) f(x-y). \]

If \( n > 0 \), then \( f(n) > 0 \) and \( f(2015) f(2016) \) is a perfect square. Find the minimum value of \( f(1) + f(2) \).

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