
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 two different planes $\alpha$ and $\beta$, and two non-coincident lines, $m$ and $n$, consider the following four propositions:  
① If $m \parallel n$ and $m \perp \alpha$, then $n \perp \alpha$.  
② If $m \perp \alpha$ and $m \perp \beta$, then $\alpha \parallel \beta$.  
③ If $m \perp \alpha$, $m \parallel n$, and $n \subset \beta$, then $\alpha \perp \beta$.  
④ If $m \parallel \alpha$ and $\alpha \cap \beta = n$, then $m \parallel n$.  
The number of incorrect propositions is.

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