Question 129
If $a_1, a_2, \dots, a_{10}$ are non-negative rational numbers with sum $6$, denote $S_k = a_k + a_{k+1} + a_{k+2} + a_{k+3}$, $1 \le k \le 7$, $M$ the largest element of the set $S = \{S_1, S_2, \dots, S_7\}$ and $m$ the smallest element of $S$. Find the largest possible $m$ and the smallest possible $M$.