
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 $f(x)=\sin x\tan x.$ Consider the following statements:  
$(1)$ The function $f(x)$ is an even function;  
$(2)$ The function $f(x)$ is increasing in the interval $\left(- \frac {\pi}{2},0\right)$;  
$(3)$ The smallest positive period of the function $f(x)$ is $2\pi$;  
$(4)$ The graph of the function $f(x)$ is symmetric about the line $x=\pi$.  
Among these statements, the correct ones are __________. (Write down all the correct statement numbers.)

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