
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 $\alpha$ and $\beta$ are two different planes in space, and lines $a$ and $b$ are two different lines in space. The following four propositions are given:  
① If $\alpha \parallel \beta$ and $a \subset \alpha$, then $a \parallel \beta$;  
② If $b \subset \beta$ and the angle formed by $a$ and $b$ is $\theta$, then the angle formed by $a$ and $\beta$ is also $\theta$;  
③ If $\alpha \perp \beta$ and $a \perp \alpha$, then $a \parallel \beta$;  
④ If $a$ and $b$ are skew lines, and $a, b \nsubseteq \alpha$, then the projections of $a$ and $b$ on $\alpha$ are two intersecting lines. Among these propositions, the correct ones are numbered as     . (Note: Write down the numbers of all propositions you think are correct)

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