
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 $a$, $b$, $c$ are three different lines, and $\alpha$, $\beta$, $\gamma$ are three different planes, then among the following propositions, the correct serial numbers are ______.

$(1)$ If $a \perp c, b \perp c$, then $a \parallel b$;  
$(2)$ If $\alpha \perp \gamma, \beta \perp \gamma$, then $\alpha \parallel \beta$;  
$(3)$ If $a \perp \alpha, b \perp \alpha$, then $a \parallel b$;  
$(4)$ If $a \perp \alpha, \alpha \perp \beta$, then $\alpha \parallel \beta$.

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