
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 $a$ and $b$ represent two different lines, and $\alpha$, $\beta$, $\gamma$ represent three different planes, consider the following propositions:  
① If $\alpha \cap \beta = a$, $b \subset \alpha$, and $a \perp b$, then $\alpha \perp \beta$;  
② If $a \subset \alpha$, and $a$ is perpendicular to any line within $\beta$, then $\alpha \perp \beta$;  
③ If $\alpha \perp \beta$, $\alpha \cap \beta = a$, and $\alpha \cap \gamma = b$, then $a \perp b$;  
④ If $a$ is not perpendicular to plane $\alpha$, then $a$ cannot be perpendicular to countless lines within plane $\alpha$;  
⑤ If $a \perp \alpha$, and $b \perp \beta$, then $\alpha \parallel \beta$.  
Among the above five propositions, the correct ones are \_\_\_\_\_\_.

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