
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 $a$, $b$, and $c$ be three different lines, and $\alpha$, $\beta$, and $\gamma$ be three different planes. Among the following four propositions, the true propositions are (write down the numbers of all true propositions):

① If $\alpha \perp \beta$ and $\beta \perp \gamma$, then $\alpha \parallel \gamma$  
② If $a \perp b$ and $b \perp c$, then $a \parallel c$ or $a \perp c$  
③ If $a \subset \alpha$, $b, c \subset \beta$, $a \perp b$, and $a \perp c$, then $\alpha \perp \beta$  
④ If $a \perp \alpha$, $b \subset \beta$, and $a \parallel b$, then $\alpha \perp \beta$

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