
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
Equilateral $\triangle PQR$ is inscribed in a circle of radius $3$. Extend $\overline{PQ}$ through $Q$ to point $S$ so that $PS=15,$ and extend $\overline{PR}$ through $R$ to point $T$ such that $PT = 9.$ Through $S,$ draw a line $l_1$ parallel to $\overline{PT},$ and through $T,$ draw a line $l_2$ parallel to $\overline{PS}.$ Let $U$ be the intersection of $l_1$ and $l_2.$ Let $V$ be the point on the circle that is collinear with $P$ and $U$ and distinct from $P.$ If the area of $\triangle QRV$ can be expressed in the form $\frac{a\sqrt{b}}{c},$ where $a, b,$ and $c$ are integers, find $a+b+c.$

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