
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  $f : \mathbb R \to \mathbb R$  satisfying the following two conditions:

(a)  $f(x + y) + f(x - y) = 2f(x)f(y)$  for all  $x, y \in \mathbb R$ , and

(b)  $\lim_{x\to \infty} f(x) = 0$ .

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