
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
An equilateral triangle $\triangle XYZ$ is inscribed in a circle of radius $3$. Extend $\overline{XY}$ through $Y$ to point $P$ so that $XP = 15$, and extend $\overline{XZ}$ through $Z$ to point $Q$ so that $XQ = 13$. Through $P$, draw a line $m_1$ parallel to $\overline{XQ}$, and through $Q$, draw a line $m_2$ parallel to $\overline{XP}$. Let $R$ be the intersection of $m_1$ and $m_2$. Let $S$ be the point on the circle such that the segment $\overline{RS}$ is perpendicular to segment $\overline{XR}$. Also, given that the area of $\triangle YZS$ can be expressed as $\frac{p\sqrt{q}}{r}$, find $p+q+r$, where $p, q, r$ are integers, $p$ and $r$ are coprime, and $q$ is square-free.

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