Avoid the Annoying Circular Objects

Geometry Level 3

You are currently located on point (0,0)(0,0) and you want to get on point (54,18)(54,18). However, there are some annoying circular objects in the way! They are defined by x2+y218x18y+81=0x^2+y^2-18x-18y+81=0 x2+y290x18y+2025=0x^2+y^2-90x-18y+2025=0

If you cannot walk through these annoying circular objects, then the shortest possible path possible to point (54,18)(54,18) can be expressed as a+bc+dπa+b\sqrt{c}+d\pi for positive integers a,b,c,da,b,c,d with cc square-free. What is a+b+c+da+b+c+d?

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