
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
Suppose points  $F_1, F_2$  are the left and right foci of the ellipse  $\frac{x^2}{16}+\frac{y^2}{4}=1$  respectively, and point  $P$  is on line  $l:$ ,  $x-\sqrt{3} y+8+2\sqrt{3}=0$ . Find the value of ratio  $\frac{|PF_1|}{|PF_2|}$  when  $\angle F_1PF_2$  reaches its maximum value.

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