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\includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-2_526_659_1336_742} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) on \(O A\) is such that \(B C\) is perpendicular to \(O A\). The point \(D\) is on \(B C\) and the circular arc \(A D\) has centre \(C\).

1. Find \(A C\) in terms of \(r\) and \(\theta\).
2. Find the perimeter of the shaded region \(A B D\) when \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 4\), giving your answer as an exact value.