
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
Rectangle $EFGH$ and a semicircle with diameter $EF$ are coplanar and have nonoverlapping interiors. Let $\mathcal{S}$ denote the region enclosed by the semicircle and the rectangle. Line $m$ meets the semicircle, segment $EF$, and segment $GH$ at distinct points $P$, $V$, and $S$, respectively. Line $m$ divides region $\mathcal{S}$ into two regions with areas in the ratio $2: 3$. Suppose that $EV = 100$, $EP = 150$, and $VF = 200$. Then $HE$ can be represented as $k\sqrt {l}$, where $k$ and $l$ are positive integers and $l$ is not divisible by the square of any prime. Find $k + l$.

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