
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 vertex $V_1$ at the origin and focus $F_1$. Two points $C$ and $D$ on $\mathcal P$ satisfy $\angle CV_1D = 90^\circ$. The locus of the midpoint of $\overline{CD}$ forms another parabola $\mathcal R$. Let $V_3$ and $F_3$ be the vertex and focus of $\mathcal R$, respectively. Determine the ratio $\frac{F_1F_3}{V_1V_3}$ if $\mathcal P$ is defined by $y = 4x^2$.

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