
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 that $a$ and $b$ are lines, and $\alpha$, $\beta$, $\gamma$ are planes, there are the following four propositions:  
$(1)$ If $a\parallel\alpha$ and $b\parallel\beta$, then $a\parallel b$;  
$(2)$ If $a\perp\gamma$ and $b\perp\gamma$, then $a\parallel b$;  
$(3)$ If $a\parallel b$ and $b\subset\alpha$, then $a\parallel\alpha$;  
$(4)$ If $a\perp b$ and $a\perp\alpha$, then $b\parallel\alpha$;  
Among these propositions, the correct one(s) is/are __________.

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