
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
Let $\alpha$, $\beta$, $\gamma$ represent three different planes, and $a$, $b$, $c$ represent three different lines. Consider the following five propositions:  
(1) If $a \parallel \alpha$, $b \parallel \beta$, and $a \parallel b$, then $\alpha \parallel \beta$;  
(2) If $a \parallel \alpha$, $b \parallel \alpha$, $\beta \cap \alpha = c$, $a \subset \beta$, $b \subset \beta$, then $a \parallel b$;  
(3) If $a \perp b$, $a \perp c$, $b \subset \alpha$, $c \subset \alpha$ $\Rightarrow$ $a \perp \alpha$;  
(4) If $\alpha \perp \gamma$, $\beta \perp \gamma$, then $\alpha \parallel \beta$ or $\alpha \perp \beta$;  
(5) If the projections of $a$ and $b$ within plane $\alpha$ are perpendicular to each other, then $a \perp b$.  
The correct propositions are numbered as follows:

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