
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 the function $f(x) = \cos x \sin 2x$, which of the following statements are correct? Fill in the correct serial numbers.

① $f(x)$ is both an odd function and a periodic function;  
② The graph of $y=f(x)$ is symmetric about the line $x= \frac{\pi}{2}$;  
③ The maximum value of $f(x)$ is $\frac{4\sqrt{3}}{9}$;  
④ $y=f(x)$ is increasing on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]$.

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