
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
a) Two parallel lines \(\ell_{1}\) and \(\ell_{2}\) touch a circle \(\omega_{1}\) with center \(O_{1}\) at points \(A\) and \(B\) respectively. Circle \(\omega_{2}\) with center \(O_{2}\) is tangent to line \(\ell_{1}\) at point \(D\), intersects line \(\ell_{2}\) at points \(B\) and \(E\), and intersects the circle \(\omega_{1}\) again at point \(C\) (with \(O_{2}\) lying 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 4. Find the ratio of the radii of circles \(\omega_{2}\) and \(\omega_{1}\).

b) Find these radii if it is additionally known that \(B D = 4\).

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