
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 defined by the equation $y = -x^2$. Let $V_1$ and $F_1$ be its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal{P}$ so that $\angle AV_1 B = 90^\circ$. Let $\mathcal{Q}$ be the locus of the midpoint of $\overline{AB}$. $\mathcal{Q}$ turns out to be a parabola, and let $V_2$ and $F_2$ denote its vertex and focus, respectively. Determine the ratio $\frac{F_1F_2}{V_1V_2}$.

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