On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
Let $O$ be the origin. When the scanning rod stops, the positions of the black and white ends are $A ^ { \prime }$ and $B ^ { \prime }$ respectively. In the diagram area of the answer sheet, use hatching to indicate the region $R$ swept by the scanning rod; and in the solution area, find $\cos \angle O A ^ { \prime } B ^ { \prime }$ and the polar coordinates of point $A ^ { \prime }$ . (Non-multiple choice question, 6 points)