
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

How many functions \( f: \mathbb{Z} \rightarrow \mathbb{R} \) satisfy the following three properties?

(a) \( f(1) = 1 \);

(b) For all \( m, n \in \mathbb{Z} \), \( f(m)^{2} - f(n)^{2} = f(m+n) f(m-n) \);

(c) For all \( n \in \mathbb{Z} \), \( f(n) = f(n+2013) \).

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