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 \(AB + BC + CA = p\sqrt{q},\) for positive integers \(p\) and \(q\) such that \(q\) is not divisible by the square of any prime, find \(p+q.\)

**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 next problem here.

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