
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 \( M \) be the set of functions \( f: \mathbf{Z} \rightarrow \mathbf{R} \) that satisfy the condition \( f(0) \neq 0 \) and the identity

\[
f(n) f(m) \equiv f(n+m)+f(n-m), \quad n, m \in \mathbf{Z}
\]

Find: 
a) all functions \( f(n) \in M \) for which \( f(1)=5 / 2 \); 
b) all functions \( f(n) \in M \) for which \( f(1)=\sqrt{3} \).

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