The diagram shows two circles with diameters **314** and **358** which share a common tangent.

Both circles are tangent to the two smaller circles in the middle with diameters **265** and **159**.

The length of the tangent in its simplest form can be expressed as \[\dfrac{\sqrt{15}}{4}(a\sqrt{b}+\sqrt{c}),\] where \(a,b\) and \(c\) are positive integers with \(b,c\) square-free. Find \(a+b+c\).

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