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Adam has a circle of radius $1$ centered at the origin.

- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.

- Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis.

- Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle.

What is the product of the lengths of all $18$ segments Adam drew?

- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.

- Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis.

- Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping when he reaches the boundary of the circle.

What is the product of the lengths of all $18$ segments Adam drew?