
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 a hyperbola with eccentricity $e$ and an ellipse with eccentricity $\frac{\sqrt{2}}{2}$ share the same foci $F_{1}$ and $F_{2}$. If $P$ is a common point of the two curves and $\angle F_{1}PF_{2}=60^{\circ}$, then $e=$ ______.

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