
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
In a configuration similar to the original, rectangle $PQRS$ and a semicircle with diameter $PQ$ are coplanar with nonoverlapping interiors. Let $\mathcal{S}$ denote the region enclosed by the semicircle and the rectangle. Line $m$ meets the semicircle, segment $PQ$, and segment $RS$ at distinct points $V$, $W$, and $Z$, respectively. Line $m$ divides region $\mathcal{S}$ into two regions with areas in the ratio $2:3$. Given that $PW = 72$, $PV = 108$, and $WQ = 144$, find the length of $PR$, which can be expressed as $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a + b$.

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