
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
Let $e_1$ and $e_2$ be the eccentricities of an ellipse and a hyperbola, respectively, both having common foci $F_1$ and $F_2$. Let $P$ be a common point of the two curves, satisfying $|\overrightarrow{PF_1} + \overrightarrow{PF_2}| = |\overrightarrow{F_1F_2}|$, then $\frac{e_1e_2}{\sqrt{e_1^2 + e_2^2}}$ = ______.

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