As shown above, there is a half circle whose diameter is segment \(\text{AB}\) and radius is \(5\).

Let \(\text{P}\) be on the half circle.

A line segment that has midpoints of chord \(\text{AP}\) and arc \(\text{AP}\) as its ends is circle \(O_1\)'s diameter. Similarly, a line segment that has midpoints of chord \(\text{BP}\) and arc \(\text{BP}\) as its ends is circle \(O_2\)'s diameter.

Circle \(O\) is an inscribed circle of \(\triangle\text{ABP}\).

The minimum of the sum of the areas of \(O\), \(O_1\), and \(O_2\) is \(S\).

Find the value of \(\dfrac{144S}{\pi}\).

*This problem is a part of <Grade 10 CSAT Mock test> series.*

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