
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 ($C_1$) and a hyperbola ($C_2$) with the center at the origin and a common focus on the $x$-axis, let $F_1$ and $F_2$ be the left and right foci, respectively. Point $P$ is the intersection of $C_1$ and $C_2$ in the first quadrant, and $\triangle PF_1F_2$ is an isosceles triangle with base $PF_1$. If the eccentricities of the ellipse and hyperbola are $e_1$ and $e_2$, respectively, find the range of $e_1 \cdot e_2$.

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