
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 and a hyperbola share the same foci $F_1$, $F_2$, and $P$ is one of their intersection points, $\angle F_1 P F_2 = 60^\circ$. Let the eccentricities of the ellipse and hyperbola be $e_1$ and $e_2$ respectively. Find the minimum value of $e_1^2 + e_2^2$.

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