
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
Given a function $f(x)$ defined on $\mathbb{R}$, for any $x, y \in \mathbb{R}$, it holds that $f(x+y) + f(x-y) = 2f(x)f(y)$, and $f(x)$ is not identically zero. Which of the following conclusions are correct?

① $f(0)=0$  
② $f(0)=1$  
③ $f(0)=0$ or $f(0)=1$  
④ The function $f(x)$ is an even function  
⑤ If there exists a real number $a \neq 0$ such that $f(a)=0$, then $f(x)$ is a periodic function and $2a$ is one of its periods.

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