7 The diagram shows the sketch of a curve \(C _ { 1 }\).
\includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve \(C _ { 1 }\) is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Find the area of the region bounded by the curve \(C _ { 1 }\).
2. The curve \(C _ { 2 }\) whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve \(C _ { 1 }\) at the pole \(O\) and at the point \(A\). The straight line drawn through \(A\) parallel to the initial line intersects \(C _ { 1 }\) again at the point \(B\).
   1. Find the polar coordinates of \(A\).
   2. Show that the length of \(O B\) is \(\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )\).
   3. Find the length of \(A B\), giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141}
      \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}