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Geometry Level 5

A sphere of radius \(1\) is tangent to the plane \(\Pi\) at the point \(A.\) A line \(l,\) that makes an angle \(\phi\) with \(\Pi,\) intersects \(\Pi\) at a point \(C.\) It is tangent to the sphere at a point \(B.\) The length of \(AC\) is \(2\) and \(\tan \phi = \frac{5}{12}.\) The square of the length of \(AB\) can be written as \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers. What is \(a+b?\)

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