Pi Day Challenge 2016 - Adagio

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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Happy Pi Day 2016.

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