
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 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 also intersects $\Omega$ at points $D$ and $E$ (with point $D$ lying 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 (with point $F$ lying between points $P$ and $H$). It is known that $BC = 5$, $DH = HC = 2$. Find the length of segment $HP$ and the radii of both circles.

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