
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
The circles \(\Omega\) and \(\omega\) touch each other externally at point \(F\), and their common external tangent touches the circles \(\Omega\) and \(\omega\) at points \(A\) and \(B\) respectively. A line \(\ell\) passes through point \(B\), intersects circle \(\omega\) again at point \(C\), and intersects \(\Omega\) at points \(D\) and \(E\) (point \(D\) is located between \(C\) and \(E\)). The common tangent of the circles passing through point \(F\) intersects lines \(AB\) and \(BE\) at points \(P\) and \(H\) respectively (point \(F\) lies between points \(P\) and \(H\)). It is known that \(DE = \frac{10}{3}\), \(DH = HC = \frac{1}{9}\). Determine the length of segment \(HP\) and the radii of both circles.

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