
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
1. If line $a$ is perpendicular to two intersecting lines within plane $\alpha$, then $a \perp \alpha$;  
2. If line $a$ is perpendicular to countless lines within plane $\alpha$, then $a \perp \alpha$;  
3. If line $b \parallel \beta$, then $b$ is parallel to all lines within plane $\beta$;  
4. If $a \subset \alpha$, $b \subset \beta$, and $a \perp b$, then $\beta \perp \alpha$;  
5. If $a \subset \alpha$, $b \subset \beta$, and $\beta \parallel \alpha$, then $a \parallel b$;  
6. If $b \subset \beta$, and $b \perp \alpha$, then $\beta \perp \alpha$;  
Among these, the correct ones are (fill in the numbers only).

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