
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 $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 13$ and $CD = 21$. Point $P$ is on segment $AB$ with $AP = 7$, and $Q$ is on segment $CD$ with $CQ = 8$. The line through $P$ and $Q$ intersects the circle at points $X$ and $Y$. If $PQ = 30$, find the length of segment $XY$.

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