Tempting Circle

Geometry Level 2

In a circle with center \(O\), a chord \(AB\) is drawn such that \(\angle AOB = 120^\circ\). Radius \(AO=10\). A circle is drawn in the major arc such that it's radius is maximum, with center E, it touches the larger circle at point X as shown in figure.

The area of the shaded region can be expressed as \(\dfrac{a\pi +b\sqrt{3}}{c}\) for integers \(a,b,c\) with \(\gcd(a,c)=1\) What is the value of \(a+b+c\)?

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