A \(2 \times 2\) square is cut into two pieces, \(A,\) which has area 3, and \(B,\) which is a square of area \(1,\) as shown in the figure below.

A \(2 \times 8\) grid is to be tiled with various pieces having shape \(A\) or \(B.\) Pieces of shape \(A\) can be rotated by \(0, \frac{\pi}{2}, \pi, \mbox{ or } \frac{3\pi}{2}\) before being placed. Subject to the condition that three pieces of shape \(B\) cannot occur on the grid in such a way that they could be replaced by a piece of shape \(A\), how many different tilings are there of the grid?

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