Let \(A_1 = (0,0), A_2=(1,0), A_3=(2,0), A_4=(3,0)\) and \(B_1 = (0,1), B_2=(1,1), B_3=(2,1), B_4=(3,1).\) Let \(L\) be the set of line segments \(A_iB_j\) where \(i\) and \(j\) are integers with \(1 \le i, j \le 4.\) (\(i\) and \(j\) do not need to be distinct.) Let \(S\) be the set of points in the interior of rectangle \(A_1A_4B_4B_1\) that are the intersection of two distinct line segments in \(L.\)
Find the number of ways to choose points \(A_i,A_j,P_1\) and \(P_2,\) where \(i\) and \(j\) are integers with \(1 \le i < j \le 4,\) \(P_1\) and \(P_2\) are points in \(S\) (the order of \(P_1\) and \(P_2\) does not matter), and \(A_iP_1P_2A_j\) forms a trapezoid with \(P_1P_2 \parallel A_iA_j.\)