Tangent bisectors

Geometry Level 5

In a triangle ABCABC, BKBK is an angle bisector. A circle with radius 53\frac{5}{3} passes through the vertex BB, intersects ABAB at a point L,L, and is tangent to ACAC at KK. It is known that the length of ACAC is 33,3\sqrt{3}, and the ratio of the lengths AK|AK| to BL|BL| is 6:56:5. The area of the triangle ABCABC can be written as abc \frac{a\sqrt{b}}{c} , where aa and cc are coprime positive integers, and bb is not divisible by the square of any prime. What is the value of a+b+ca+b+c?

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