
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 $y^{2}=4x$ with focus $F$, a line $l$ passing through $F$ intersects the parabola and the circle $(x-1)^{2}+y^{2}=\frac{1}{4}$ at four points, $A, B, C, D$ from top to bottom. Find the minimum value of $|AB|+4|CD|$.

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