
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 $f(x)=\sin (2x- \frac {\pi}{4})$, among the following four propositions:  
$(1)f(x)$'s minimum positive period is $4\pi$;  
$(2)f(x)$'s graph can be obtained by shifting the graph of $g(x)=\sin 2x$ to the right by $\frac {\pi}{4}$ units;  
$(3)$ If $x_{1}\neq x_{2}$ and $f(x_{1})=f(x_{2})=-1$, then $x_{1}-x_{2}=k\pi(k\in\mathbb{Z}$, and $k\neq 0)$;  
$(4)$ The line $x=- \frac {\pi}{8}$ is a symmetry axis of the graph of the function $f(x)$.  
The correct proposition numbers are __________ (Fill in the numbers of the propositions you think are correct).

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