
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 ellipse \(\frac{x^{2}}{1}+y^{2}=1\) with left and right foci \(F_{1}\) and \(F_{2}\), and point \(P\) on the line \(l: x+2 \sqrt{3} y-4 \sqrt{3}=0\). Find the value of \(\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}\) when \(\angle F_{1} P F_{2}\) is maximized.

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