The diagram above shows a big red circle with diameter **314159**,
tangent to a cyan circle with diameter **26535** and a green circle with diameter **23846**,
which are tangent to each other and also to a purple circle with diameter **89793**.

The length of the pink line segment between the two points of tangency can be expressed in its simplest form as \[\dfrac{2\sqrt{105458935}}{50381}\big(a\sqrt{b}+\sqrt{c}\big),\] where \(a,b\) and \(c\) are positive integers with \(b,c\) square-free. Find \(a+b+c\).

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