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 $a+b+c$.