
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
Let there be planes $\alpha$ and $\beta$, lines $a$ and $b$, and sets $A=\{\text{planes perpendicular to }\alpha\}$, $B=\{\text{planes perpendicular to }\beta\}$, $M=\{\text{lines perpendicular to }a\}$, $N=\{\text{lines perpendicular to }b\}$. Consider the following propositions:  
① If $A\cap B \neq \emptyset$, then $\alpha \parallel \beta$;  
② If $\alpha \parallel \beta$, then $A=B$;  
③ If $a$ and $b$ are skew lines, then $M\cap N = \emptyset$;  
④ If $a$ and $b$ intersect, then $M=N$;  
Among these, the incorrect propositions are.

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