Hey ant, I just want a word for you: the word is stop!

Geometry Level 4

In the figure shown above \(ABC\) is a right triangle with \(\angle B = 90^\circ\), \(AB=10-\pi\) and \(BC=6\). There is a semi-cylinder with radius \(1\) and its axis is perpendicular to \(AB\). That little ant in point \(A\) must scape to point \(C\) soon! What is the length of shortest way for him to going to his nest in point \(C\)?

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