
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 the parabola \( C: y^2 = 4x \) with focus \( F \), points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) lie on the parabola \( C \) such that \( x_1 + x_2 = 4 \). Let \( l \) be the line containing the common chord of the two circles with diameters \( AF \) and \( BF \). The line \( l \) intersects the parabola \( C \) at points \( M \) and \( N \). Find the range of \( |MN| \).

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