
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 $f: \mathbb{Z} \rightarrow \mathbb{R}$ that satisfy \( f(0) \neq 0 \) and
\[ f(n) f(m) \equiv f(n + m) + f(n \lfloor m), \quad n, m \in \mathbb{Z} \). 
Find:

(1) All functions $f(n) \in M$ that satisfy $f(1) = \frac{5}{2}$;

(2) All functions $f(n) \in M$ that satisfy $f(1) = \sqrt{3}$.

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