
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 $P$ so that $XP=10,$ and extend $\overline{XZ}$ through $Z$ to point $Q$ so that $XQ = 7.$ Through $P,$ draw a line $l_1$ parallel to $\overline{XQ},$ and through $Q,$ draw a line $l_2$ parallel to $\overline{XP}.$ Let $R$ be the intersection of $l_1$ and $l_2.$ Let $S$ be the point on the circle that is collinear with $X$ and $R$ and distinct from $X.$ Given that the area of $\triangle ZYS$ can be expressed in the form $\frac{a\sqrt{b}}{c},$ where $a, b,$ and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime, find $a+b+c.$

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