[Central Angle. Arc Length and Circumference]

On the coordinate plane ($x ; y$), a circle with radius 4 and center at the origin is drawn. The line given by the equation $y=\sqrt{3} x-4$ intersects it at points $A$ and $B$. Find the sum of the lengths of segment $A B$ and the larger arc $A B$.
Solve the system of equations to find the coordinates of the points of intersection between the line and the circle. Then, using the cosine theorem, find the angle between the radii of the circle drawn to the found points of intersection.

## Answer

$\frac{16 \pi}{3}+4 \sqrt{3}$