
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 the following four propositions:
① If $\alpha \perp \beta$, then there must exist a line within $\alpha$ that is parallel to plane $\beta$.
② If $\alpha \perp \beta$, then all lines within $\alpha$ are perpendicular to plane $\beta$.
③ If $\alpha \perp \gamma$, $\beta \perp \gamma$, and $\alpha \cap \beta = l$, then $l \perp \gamma$.
④ If $\alpha \parallel \beta$, $\beta \parallel \gamma$, and $m \perp \alpha$, then $m \perp \gamma$.
Among these, the numbers of the true propositions are.

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