
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
Consider the function $f(x) = x \cdot \arcsin(x)$. The following statements are given:  
① The domain of $f(x)$ is $\mathbb{R}$;  
② $f(x)$ is an even function;  
③ $f(x)$ is an increasing function within its domain;  
④ The maximum value of $f(x)$ is $\frac{\pi}{2}$ and the minimum value is $0$;  
Identify the correct statement(s). __(Write down all the numbers of the propositions you think are correct)__.

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