$(20$ points$)$ Consider the gene assembly paper Simple Operations for Gene Assembly and refer to Section $4$

and the formulas for $sh$ and $sd$ on p$.100.$

$($a$)$ For the signed permutation ${}_1=\frac{1}{b}ar(3)\frac{42}{b}ar(5),$ use the sorting strategy $s{h}_5$@$s{d}_2$@$s{h}_3$ to sort the permutation

${}_1.$ Show every step of the sorting $($by showing the formula used at each step and the result of applying the

formula$)$ and state the sorted permutation.

$($b$)$ For the signed permutation ${}_2=\frac{1}{b}ar(2)\frac{43}{b}ar(5)6,$ find a sorting strategy, state it$,$ and then use it to sort the

permutation. Show every step of the sorting $($by showing the formula used and the result of applying the

formula$)$ and state the sorted permutation.