Understanding Convex Octagons

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).