
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 a function $f(x)$ defined on $\mathbb{R}$, for any real numbers $x$ and $y$, it satisfies $f(x+y)+f(x-y)=2f(x)\cos y$, and $f(0)=0$, $f\left( \frac {\pi }{2}\right)=1$. Consider the following statements:  
$(1) f\left( \frac {\pi }{4}\right)= \frac {1}{2}$  
$(2) f(x)$ is an odd function  
$(3) f(x)$ is a periodic function  
$(4) f(x)$ is monotonic in $(0,\pi)$  
The correct statements are __________. (Fill in all the correct statement numbers).

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