
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
In an equilateral triangle $\triangle PQR$ 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 $T$ so 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 $P$. Given that the area of $\triangle QRV$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p, q, and r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.

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