$2.(50)[$Counting inversion: algorithm tracing$]$ Consider the counting inversion algorithm Sort$-$andCount studied in class and shown below.

$```$

Sort$-$and$-$Count$($L$)\{$

if list L has one element

return $0$ and the list L

Divide the list into two halves A and B

$($r A

$(\}\backslash$mp@subsup$\{$r$\}\{\backslash$textrm$\{$B$\}\}\{\},$B$)\backslash$mathrm$\{$ B$)\}\backslash$leftarrow$\backslash$mathrm$\{$ Sort$-$and$-$Count $($B$)$

$(\}\backslash$mp@subsup$\{$r$\}\{$AB$\}\{\},$L$)\backslash$leftarrow$\backslash$mathrm$\{$ Merge$-$and$-$Count$($A$,$ B$)$

return r $=$ r A $+$ r B $+$ r

$\}$

$```$

Run this algorithm against the input file shown below, and $($a$)$ show the recursion tree during the divide phase and the conquer $\backslash$& combine phase, respectively, and $($b$)$ show each returned result as the pair of the inversion count and the sorted result $($e$.$g$.,"\backslash ((1,\backslash \{4,6\backslash \})\backslash )")$ exactly in the order of execution. If the input array has an odd number of elments, the left half contains one fewer element than the right half. See the example recursion trees below.

Input file: $\backslash (\backslash \{7,3,11,12,9,6,2,10,8,4,5,1\backslash \}\backslash )$

Example recursion tree for an input file $\backslash (\backslash \{6,4,2,3\backslash \}\backslash )$

Divide