
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 given setup, a rectangle $ABCD$ is adjacent to a semicircle with diameter $AB$. Define region $\mathcal{S}$ encompassed by the rectangle and the semicircle. A line $m$ intersects the semicircle, line segment $AB$, and line segment $CD$ at points $M$, $V$, and $S$, respectively. The line $m$ divides the region $\mathcal{S}$ into two parts with areas in the ratio $3:4$. Given that $AV = 80$ and $VM = 120$ with $VB = 200$, calculate the length of $DA$, denoted as $p\sqrt{q}$, where $p$ and $q$ are integers, and $q$ is not divisible by the square of any prime. Find $p + q$.

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