In a $33$ transportation problem, let $$be the amount shipped from source $$to destination $,$ and let $$be the corresponding transportation cost per unit. The amounts of supply at sources $1,2,$ and $3$ are $25,40,$ and $55$ units, respectively, and the demands at destinations $1,2,$ and $3$ are $30,40,$ and $50$ units, respectively. Assume that the starting northwest$-$corner solution is optimal and that the associated values of the multipliers are given as $$u$1=0,$ u$2=4,$ u$3=6,$ v$1=2,$ v$2=5,$ v$3=7.$

Determine the smallest value of c$12,$ c$13,$ c$23,$ and c$31.$