
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 lines $\alpha$, $\beta$, $\gamma$ respectively, and they are in a plane, then among the following propositions, the correct proposition numbers are:

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

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