
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
Let \( \mathcal{P'} \) denote a parabola defined by the equation \( y = 2x^2 \), with vertex \( V_1' \) and focus \( F_1' \). Let \( C \) and \( D \) be points on \( \mathcal{P'} \) such that \( \angle CV_1'D = 90^\circ \). Determine the locus \( \mathcal{Q'} \) of the midpoint of the segment \( \overline{CD} \), and let \( V_2' \) and \( F_2' \) be the vertex and focus of \( \mathcal{Q'} \), respectively. Compute the ratio \( \frac{F_1'F_2'}{V_1'V_2'} \).

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