
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 $\mathbb{R}$, and for any $x \in \mathbb{R}$, it satisfies $f(x+1) = f(x-1)$. Given that when $x \in [0, 1)$, $f(x) = \log_{0.5}(1-x)$. Determine the correctness of the following statements:
① 2 is a period of the function $f(x)$;
② $f(x)$ is increasing on the interval $(1, 2)$ and decreasing on the interval $(2, 3)$;
③ The maximum value of $f(x)$ is 1, and the minimum value is 0;
④ When $x \in (3, 4)$, $f(x) = \log_{0.5}(x-3)$.

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