
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
Two parallel lines $\ell_{1}$ and $\ell_{2}$ are tangent to a circle $\omega_{1}$ centered at $O_{1}$ at points $A$ and $B$, respectively. A circle $\omega_{2}$ centered at $O_{2}$ is tangent to line $\ell_{1}$ at point $D$, intersects line $\ell_{2}$ at points $B$ and $E$, and intersects circle $\omega_{1}$ again at point $C$ (with $O_{2}$ located between lines $\ell_{1}$ and $\ell_{2}$). It is known that the ratio of the area of quadrilateral $B O_{1} C O_{2}$ to the area of triangle $O_{2} B E$ is $\frac{5}{4}$. Find the ratio of the radii of circles $\omega_{2}$ and $\omega_{1}$.

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