
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
If $\alpha$ and $\beta$ are two intersecting planes, then among the following propositions, the numbers of the true propositions are (write down the numbers of all true propositions):

① If line $m \perpendicular \alpha$, then in plane $\beta$, there definitely does not exist a line parallel to line $m$.

② If line $m \perpendicular \alpha$, then in plane $\beta$, there definitely exist infinitely many lines perpendicular to line $m$.

③ If line $m \subset \alpha$, then in plane $\beta$, there does not necessarily exist a line perpendicular to line $m$.

④ If line $m \subset \alpha$, then in plane $\beta$, there definitely exists a line perpendicular to line $m$.

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