
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
Find all functions \( f: \mathbf{Z} \rightarrow \mathbf{R} \) such that for all integers \( x \) and \( y \), 
$$
\begin{array}{l}
f(x) f(y) = f(x+y) + f(x-y), \\
f(1) = \frac{5}{2}.
\end{array}
$$

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