
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 an ellipse and a hyperbola with common foci centered at the origin, and their left and right foci are respectively $F_{1}$ and $F_{2}$, these two curves intersect at point $P$ in the first quadrant, and $\triangle PF_{1}F_{2}$ is an isosceles triangle with $PF_{1}$ as its base. If $|PF_{1}|=10$, and the eccentricities of the ellipse and hyperbola are respectively $e_{1}$ and $e_{2}$, determine the range of values for $e_{1} \cdot e_{2}$.

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