
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 three planes $\alpha$, $\beta$, and $\gamma$, where $\beta \perp \gamma$ and $\alpha$ intersects with $\gamma$ but not perpendicularly, the lines a, b, c are within planes $\alpha$, $\beta$, $\gamma$ respectively. Among the following propositions: ① Any line $b \subset \beta$ is perpendicular to $\gamma$; ② Any line $b \subset \beta$ is parallel to $\gamma$; ③ There exists a line $a \subset \alpha$ that is perpendicular to $\gamma$; ④ There exists a line $a \subset \alpha$ that is parallel to $\gamma$; ⑤ Any line $c \subset \gamma$ is parallel to $\alpha$; ⑥ There exists a line $c \subset \gamma$ that is perpendicular to $\beta$. The sequence numbers of true propositions are ___.

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