
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 rectangle $EFGH$, a semicircle with diameter $EF$ is coplanar and shares no overlapping interiors with the rectangle. Let $\mathcal{S}$ denote the region enclosed by the semicircle and the rectangle. A line $\ell$ meets the semicircle, segment $EF$, and segment $GH$ at distinct points $P$, $V$, and $Q$, respectively. This line divides region $\mathcal{S}$ into two regions having areas in the ratio $3: 1$. If $EV = 70$, $EP = 105$, and $VF = 210$, determine the length of $EG$.

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