A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square. Such a board is called interesting if the following conditions hold: $\circ$ In all unit squares below the main diagonal, the number $0$ is written; $\circ$ Positive integers are written in all other unit squares. $\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers are actually the numbers $1,2,...,2n$ (not necessarily in that order). $a)$ Determine the largest number that can appear in a $6 \times 6$ interesting board. $b)$ Prove that there is no interesting board of dimensions $7\times 7$.