
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 that $P$, $Q$, $R$ are three distinct points on the circle $x^{2}+y^{2}-x-15=0$, and their distances to the line $l:x+\sqrt{3}y+9=0$ are $x_{1}$, $x_{2}$, and $x_{3}$ respectively, calculate the maximum value of the common ratio if $x_{1}$, $x_{2}$, and $x_{3}$ form a geometric sequence.

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