
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, with $V_1$ and $F_1$ as its vertex and focus, respectively. Let $A$ and $B$ be points on $\mathcal P$ such that $\angle AV_1 B = 60^\circ$. Let $\mathcal Q$ be the locus of the midpoint of $\overline{AB}$. It turns out that $\mathcal Q$ is also a parabola. Let $V_2$ and $F_2$ represent the vertex and focus of $\mathcal Q$, respectively. Determine the ratio $\frac{F_1F_2}{V_1V_2}$.

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