
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 \( E: y^{2} = 2x \) with focus \( F \), two lines \( l_{1} \) and \( l_{2} \) are drawn through \( F \) with their slopes' product equal to \( -\frac{1}{2} \). Line \( l_{1} \) intersects \( E \) at points \( A \) and \( C \), and line \( l_{2} \) intersects \( E \) at points \( B \) and \( D \). Find the minimum value of \( |A C| + 2|B D| \).

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