
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 $e_1$ and $e_2$ are the eccentricities of an ellipse and a hyperbola with common foci $F_1$ and $F_2$, and $P$ is a common point of the two curves such that $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = 0$, find the value of $\frac{e_1^2 + e_2^2}{(e_1 e_2)^2}$.

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