
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
Given that an ellipse $C_{1}: \frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 \ (a_{1} > b_{1} > 0)$ and a hyperbola $C_{2}: \frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 \ (a_{2} > 0, b_{2} > 0)$ share the same foci $F_{1}$ and $F_{2}$. Let point P be a common point of the two curves, and let $PF_{1} \perp PF_{2}$. Let $e_{1}$ and $e_{2}$ be the eccentricities of the curves $C_{1}$ and $C_{2}$, respectively. Find the minimum value of $9e_{1}^{2} + e_{2}^{2}$.

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