
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
$\alpha$, $\beta$, $\gamma$ are three distinct planes, and $l$, $m$, $n$ are three distinct lines. Consider the following four propositions:  
① If $\alpha \perp \gamma$ and $\beta \perp \gamma$, then $\alpha \parallel \beta$.  
② If $m \parallel \beta$ and $n \parallel \beta$, then $\alpha \parallel \beta$.  
③ If $l \subset \alpha$ and $\alpha \parallel \beta$, then $l \parallel \beta$.  
④ If $\alpha \cap \beta = \gamma$, $\beta \cap \gamma = m$, and $\gamma \cap \alpha = n$, and $l \parallel m$, then $m \parallel n$.  
The number of correct propositions is     .

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