
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
A circle is circumscribed around quadrilateral $ABCD$, tangent to $\overline{AB}$ at $P$ and to $\overline{CD}$ at $Q$. The points $P$ and $Q$ divide sides $AB$ and $CD$ into segments $AP=15$, $PB=31$, $CQ=43$, and $QD=17$. Additionally, let the radius of the circumscribing circle be $r$. Find the square of the radius $r$ of the circle.

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