
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 $y = (x-1)^2 + 1$. Let $V_1$ be the vertex and $F_1$ the focus of $\mathcal{P}$. Let $A$ and $B$ be points on $\mathcal{P}$ such that the tangent lines to $\mathcal{P}$ at $A$ and $B$ are perpendicular. Determine the locus of the midpoint of $\overline{AB}$. Let $\mathcal{Q}$ be that locus, 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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