
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 $f(x)=\sin (x-\alpha)+2\cos x$ (where $\alpha$ is a constant), the following five propositions are given:  
1. There exists $\alpha$ such that the function $f(x)$ is an even function;  
2. There exists $\alpha$ such that the function $f(x)$ is an odd function;  
3. The minimum value of the function $f(x)$ is $-3$;  
4. If the maximum value of the function $f(x)$ is $h(\alpha)$, then the maximum value of $h(\alpha)$ is $3$;  
5. When $\alpha= \frac {\pi}{6}$, $(- \frac {\pi}{3},0)$ is a symmetry center of the function $f(x)$.  
Among these, the correct proposition numbers are __________ (fill in all the correct proposition numbers).

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