
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 \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3} + y^{2} = 1 \). A line \( l \) with positive slope passes through \( F \) and intersects the ellipse \( E \) at points \( A \) and \( B \). From points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn respectively, where \( AM \perp l \) and \( BN \perp l \), intersecting the x-axis at points \( M \) and \( N \). Find the minimum value of \( |MN| \).

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