
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 $f(x)=2|\cos x|\sin x+\sin 2x$, consider the following four statements:  
$(1)$ The graph of the function $f(x)$ is symmetric about the line $x= \frac {\pi}{4}$;  
$(2)$ The function $f(x)$ is monotonically increasing in the interval $\left[- \frac {\pi}{4}, \frac {\pi}{4}\right]$;  
$(3)$ The smallest positive period of the function $f(x)$ is $\pi$;  
$(4)$ The range of the function $f(x)$ is $[-2,2]$.  
Among these, the true statements are \_\_\_\_\_\_ (Fill in the numbers of all true statements).

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