
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 $F_1$ and $F_2$ are the common foci of an ellipse and a hyperbola, and $P$ is a common point on both curves with $\angle F_1PF_2 = \frac{\pi}{3}$, find the maximum value of the sum of the reciprocals of the eccentricities of the ellipse and hyperbola.

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