Circle within a Circle

Geometry Level 3

Equilateral triangle \(ABC\) has a circumcircle \( \Gamma\) with center \(O\) and circumradius \(10\). Another circle \( \Gamma_1\) is drawn inside \( \Gamma\) such that it is tangential to radii \( OC\) and \(OB\) and circle \( \Gamma\). The radius of \( \Gamma_1\) can be expressed in the form \( a \sqrt{b} -c\), where \(a, b\) and \(c\) are positive integers, and \(b\) is not divisible by the square of any prime. What is the value of \( a + b + c \)?

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