$[$Sorting: $15\%]$ Answer two questions regarding to the stable property of sorting.

$($a$)(8\%)$ Show how to detect a sorting algorithm is stable or not. That is$,$ given an unknown sorting algorithm, can you help to write a short script to detect whether the output of the sorting algorithm produces stable outcomes.

$($b$)(7\%)$ Below is an implementation of the heapsort algorithm. Can you identify the line$($s$)$ where executing the line$($s$)$ can lead to non$-$stable sorting result, and why?

MAX$-$HEAPIFY $(A,i)$

$,1$LEFT$(i)$

$,2r$RIGHT$(i)$

$,3$ if $l$ heap$-$size A and $A[l]>A[i]$

$,4$ then largest larrl

$,5$ else largest larri

$,6$ if $r$ heap$-$size A and largest