
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 $F_1$. Let $C$ and $D$ be points on $\mathcal{P}$ such that $\angle CV_1D = 90^\circ$. Let $\mathcal{R}$ be the locus of the midpoint of $\overline{CD}$. If $\mathcal{R}$ is also a parabola with vertex $V_3$ and focus $F_3$, determine the ratio $\frac{F_1F_3}{V_1V_3}$.

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