$1$A$)$ A B$+$ tree can alternatively be established in a batched manner. Specifically, the batched algorithm sorts the tuples to be inserted based on their keys in ascending order by engaging external memory sorting and then establishing the tree layer by layer. Show the process of external memory sorting $39,97,38,81,75,-1,41,70,51,67,69,100,91,87,60,52,105,31,10,120$ using a buffer of $3$ frames and specifically, list the total number of I$/$Os used by the sorting.

$1$B$)$ Continue $(1$A$),$ show the B$+$ tree established via batched construction.