
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 $f(x)=|\cos x|\cdot\sin x$, consider the following five statements:  
$(1) f\left( \frac {2014\pi}{3}\right)=- \frac { \sqrt {3}}{4}$;  
$(2)$ If $|f(x_{1})|=|f(x_{2})|$, then $x_{1}=x_{2}+k\pi(k\in\mathbb{Z})$;  
$(3) f(x)$ is monotonically increasing in the interval $\left[- \frac {\pi}{4}, \frac {\pi}{4}\right]$;  
$(4)$ The period of the function $f(x)$ is $\pi$;  
$(5) f(x)$'s graph is centrally symmetric about the point $\left(- \frac {\pi}{2},0\right)$.  
Among these statements, the correct ones are \_\_\_\_\_\_.

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