
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 \( f(n) \) be a function defined on the set of integers \( \mathbb{Z} \), satisfying:
1. \( f(0) = 1 \), \( f(1) = 0 \);
2. \( f(m+n) + f(m-n) = 2 f(m) f(n) \), for all integers \( m \) and \( n \).

Determine \( f(n) \).

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