Connect the curve

Calculus Level 3

The figure shows the graph of $$f\left( x \right) =x+\frac { 1 }{ x }$$

The circle is tangent to the graph in 1st and 3rd quadrant and is the smallest of its kind.

The area of the circle is in the form $A\pi \left( B+\sqrt { C } \right)$ where $$A,B,C$$ are integers with $$C$$ square free.

Find $$A+B+C$$.

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