
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 that $e\_1$ and $e\_2$ are the eccentricities of an ellipse and a hyperbola with common foci $F\_1$ and $F\_2$, respectively, and $P$ is a common point of the two curves such that $| \overrightarrow{PF}\_{1}+ \overrightarrow{PF}\_{2}|=| \overrightarrow{F\_{1}F\_{2}}|$, find the value of $\frac {e\_1e\_2}{ \sqrt { e\_1^2 + e\_2^2 }}$.

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