
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 $\mathcal P$ be a parabola, and let $V_1$ and $F_1$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal P$ such that $\angle AV_1 B = 90^\circ$. Let $\mathcal Q$ be the locus of the midpoint of $\overline{AB}$. If the equation of $\mathcal P$ is $y = 2x^2$ and $V_1$ is at $(0,1)$, determine the ratio $\frac{F_1F_2}{V_1V_2}$ where $\mathcal Q$ is a parabola, and $V_2$ and $F_2$ its vertex and focus, respectively.

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