
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 propositions:
① If $a$ and $b$ are two lines, and $a \parallel b$, then $a$ is parallel to any plane that passes through $b$;
② If plane $\alpha$ is not perpendicular to plane $\beta$, then there must not exist a line in plane $\alpha$ that is perpendicular to plane $\beta$;
③ If line $a$ and line $b$ are skew lines, and line $b$ and line $c$ are skew lines, then line $a$ and line $c$ are also skew lines;
④ Given plane $\alpha \perp$ plane $\beta$, and $\alpha \cap \beta = b$, if $a \perp b$, then $a \perp$ plane $\beta$;
⑤ Given line $a \perp$ plane $\alpha$, line $b$ is in plane $\beta$, $a \parallel b$, then $\alpha \perp \beta$.
The correct propositions are numbered as follows.

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