Let's jazz up this distillation dilemma! Picture a mix of $8\%$ acetone, $61\mathrm{.}8\%$ acetic acid, and $30\mathrm{.}2\%$ acetic anhydride entering a column $-$ it's like a chemical cocktail party! The column's job? To whip up $800$ kmo$\frac{l}{h}$ of distillate, boasting $10\%$ acetone and $76\%$ acetic acid. But wait, there's more: no acetone in the bottoms stream, and $55\%$ of the overhead gets a VIP pass back as reflux. Now, here's the twist: what's the mole fraction of acetic anhydride playing it cool in the bottoms stream? Let's unravel this molecular mystery!

Round your answer to $2$ decimal places.