
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
Quadrilateral $ABCD$ is inscribed in circle $O$ with side lengths $AB=4, BC=3, CD=7$, and $DA=9$. Let $X$ and $Y$ be points on $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{3}$ and $\frac{BY}{BD} = \frac{1}{4}$. Let $E$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AD}$. Let $F$ be the intersection of line $CX$ and the line through $E$ parallel to $\overline{AC}$. Let $G$ be the point on circle $O$ other than $C$ that lies on line $CX$. What is $XF\cdot XG$?
A) $\frac{155}{4}$
B) $\frac{160}{3}$
C) $\frac{159}{3}$
D) $\frac{155}{3}$
E) $52$

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