
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
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the following two properties:

$$
\begin{gathered}
f(x+y) + f(x-y) = f(x) f(y) \\
\lim_{x \rightarrow \infty} f(x) = 0
\end{gathered}
$$

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