Let \(A_1A_2\dots A_8\) be a convex octagon such that its opposite sides are parallel. Define \(B_i\) as the intersection between \(A_{i-1}A_{i+1}\) and \(A_iA_{i+4},\) where \(A_{j+8} = A_j\) for all \(j\).

Let \(x\) be the minimum value of \(\frac{A_iA_{i+4}}{B_iB_{i+4}}\). Over all convex octagons, find the maximum possible value of \(x\) (to 3 decimal places).

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