The diagram shows part of the curve \(y = 1 - 2x - (1 - 2x)^3\) intersecting the x-axis at the origin \(O\) and at \(A \left( \frac{1}{2}, 0 \right)\). The line \(AB\) intersects the y-axis at \(B\) and has equation \(y = 1 - 2x\).

(i) Show that \(AB\) is the tangent to the curve at \(A\).

(ii) Show that the area of the shaded region can be expressed as \(\int_0^{\frac{1}{2}} (1 - 2x)^3 \, dx\).

(iii) Hence, showing all necessary working, find the area of the shaded region.