
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 $f(x + 1) = f(x - 1)$. It is known that when $x \in [0,1)$, $f(x) = \log_{0.5}(1-x)$. Determine the true statements among the following: 
1. The period of $f(x)$ is 2. 
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. For $x \in (3,4)$, $f(x) = \log_{0.5}(x-3)$. 
Identify all the correct statement numbers.

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