
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 $\triangle XYZ$ is inscribed in a circle of radius $3$. Extend $\overline{XY}$ through $Y$ to point $D$ such that $XD=17$, and extend $\overline{XZ}$ through $Z$ to point $E$ such that $XE = 19$. Draw a line $l_1$ through $D$ parallel to $\overline{XE}$, and a line $l_2$ through $E$ parallel to $\overline{XD}$. Let $F$ be the intersection of $l_1$ and $l_2$. Let $G$ be the point on the circle that is collinear with $X$ and $F$ and distinct from $X$. Determine the area of $\triangle CYG$ and express it in the form $\frac{p\sqrt{q}}{r}$ where $p, q, r$ are integers with $p$ and $r$ being relatively prime and $q$ 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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