
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 \(e_1\) and \(e_2\) be the eccentricities of an ellipse and a hyperbola, respectively, both having common foci \(F_1\) and \(F_2\). Let \(P\) be a common point of the two curves, satisfying \(\overset{→}{PF_1} \cdot \overset{→}{PF_2} = 0\). Find the value of \(\frac{1}{e_1^2} + \frac{1}{e_2^2}\).

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