Using Branch$-$and$-$Bound with advanced bounds,$$solve problems $1-12$ using the above information.$$The branching strategy is to assign one of the remaining jobs to the last available position, and the bounding strategy is to use advanced bounding presented in class.$$As for displaying nodes, we use an underscore $(\_)$ to denote a blank position and a blank space to separate jobs in subsequent positions of the sequence. For instance, $"\_\_\_\_\_"$ is the root and $"\_\_\_51"$ is a partial solution at the second level with job $1$ assigned to the $5$th position and job $5$ to the $4$th$.$ When listing children of a node, $($i$)$ list only the immediate children $($not grandchildren in lower levels of the tree$),$ and $($ii$)$ list them in ascending order of the branching job. For instance, if you list children of $"\_\_\_21,"$ the correct listing of its children is $"\_\_321,""\_\_421,"$ and finally $"\_\_521."$ Last but not least do not enter nodes with the quotation or other punctuation marks you see here.

What is the lower bound$$of the root? $[$a$]$