
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, $P$ is their common point, and $\angle F(_1)PF(_2) = \frac{\pi}{3}$, find the maximum value of the sum of the reciprocals of the eccentricities of the ellipse and the hyperbola.

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