March Break 2024 - Problem 2

Let there be a circle with radius 5 centered of O. Construct two lines that are externally tangent to the circle at A and B. The two lines intersect at a point C, and the smallest angle that the two lines create is equal to 30^\circ. The smallest possible area of quadrilateral AOBC can be expressed as a-b\sqrt{c}, where a,b and c are positive integers and c is not divisible by the square of any prime. Compute a+b+c.