Row Reduction & Rank · Superchallenge problem

Gaussian elimination with partial pivoting is applied to an invertible $6\times 6$ matrix $A$ all of whose entries lie in the interval $[-1, 1]$: at each step, a row whose entry in the pivot column has the largest absolute value is exchanged into the pivot position (no exchange if the current row already has a largest one), and then the entries below the pivot are eliminated. Over all such matrices $A$, find the largest possible value of $|u_{66}|$, the absolute value of the last pivot.