
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
We define a pair of "related curves" as an ellipse and a hyperbola that share the same foci and have reciprocal eccentricities. Given that $F\_1$ and $F\_2$ are the foci of such a pair of related curves, and $P$ is their intersection point in the first quadrant, find the eccentricity of the ellipse when $\angle F\_1PF\_2 = 60^\circ$.

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