
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 $y = 2x^2$, with vertex $V_1$ and focus $F_1$. Let $A = (a, 2a^2)$ and $B = (b, 2b^2)$ be two points on $\mathcal P$ such that $\angle AV_1 B = 120^\circ$. Find the locus $\mathcal Q$ of the midpoint $M$ of $\overline{AB}$, and let $V_2$ and $F_2$ be 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}
user