
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
Let pentagon $ABCDE$ be inscribed in circle $O$ and have side lengths $AB=4, BC=3, CD=7, DE=5$, and $EA=9$. Points $X$ and $Y$ are on line segment $\overline{BD}$ such that $\frac{DX}{BD} = \frac{1}{3}$ and $\frac{BY}{BD} = \frac{1}{5}$. Let $F$ be the intersection of line $AX$ and the line through $Y$ parallel to $\overline{AE}$. Let $G$ be the intersection of line $CX$ and the line through $F$ parallel to $\overline{CD}$. Let $H$ be the point on circle $O$ other than $D$ that lies on line $CX$. Determine the product $XG \cdot XH$.
A) 15
B) 18
C) 21
D) 22
E) 25

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