
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 a standard ellipse and a standard hyperbola with foci on the x-axis and common foci $F_1$ and $F_2$, $O$ is the coordinate origin, and $P$ is a common point on both curves satisfying $2|\overrightarrow {OP}|=|\overrightarrow {F_{1}F_{2}}|$, determine 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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