Lines \(l, m, n\) in the same plane are such that \(l \cap m = A,\) \(m \cap n = B,\) \(l \cap n = C,\) and \(A \neq B \neq C.\) The three circles that are tangent to \(l, m,\) and \(n\) but not in the interior of \(ABC\) have radii \(3, 4,\) and \(5.\) If the area of \(ABC\) can be represented by

\[\frac{p\sqrt{q}}{r},\]

where \(p, q\) and \(r\) are positive integers such that \(p\) and \(r\) are relatively prime and \(q\) is not divisible by the square of any prime, find \(p + q + r.\)

**Notes:**

Assume that such a configuration described in the problem is possible.

\(a \cap b\) means the intersection of lines \(a\) and \(b.\)

See the first problem here.

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