
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
Among the following propositions in space, the correct ones are: (Note: Fill in the numbers of the propositions you think are correct)

① If two lines $a$ and $b$ are parallel to line $l$, then $a \parallel b$  
② If line $a$ is parallel to a line $b$ within plane $\beta$, then $a \parallel \beta$  
③ If line $a$ is perpendicular to two lines $b$ and $c$ within plane $\beta$, then $a \perp \beta$  
④ If a line $a$ within plane $\beta$ is perpendicular to plane $y$, then $\beta \perp y$

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