
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 that share a common focus, with the center at the origin of the coordinate system and the left and right foci being $({F}_1, {F}_2)$, the intersection point of these two curves in the first quadrant is denoted by $P$. $\triangle P{F}_1{F}_2$ is an isosceles triangle with $PF_1$ as the base. If $|PF_1| = 10$, determine the range of $({e}_1 \cdot {e}_2)$.

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