
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 $f(x)$ be an even function defined on $\mathbb{R}$, which satisfies $f(x+1) = f(x-1)$ for any $x \in \mathbb{R}$. If $f(x) = 2^{x-1}$ for $x \in [0,1]$, then determine the correctness of the following statements:

(1) 2 is a period of the function $f(x)$.
(2) The function $f(x)$ is increasing on the interval (2, 3).
(3) The maximum value of the function $f(x)$ is 1, and the minimum value is 0.
(4) The line $x=2$ is an axis of symmetry for the function $f(x)$.

Identify the correct statements.

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