
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 the ellipse $C\_1$ and the hyperbola $C\_2$ share the same foci and their eccentricities are reciprocals of each other. $F\_1$, $F\_2$ are their common foci, and $P$ is the intersection point of the ellipse and hyperbola in the first quadrant. When $∠F\_1PF\_2=60^{\circ}$, calculate the eccentricity of the ellipse $C\_1$.

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