Problem $1.(10$ points$)$ Given an array of characters A$[1.\mathrm{.}7]=$ BFACEGD, perform Heapsort to sort A into non$-$decreasing order $($ABCDEFG$).$ Show the updated array after each swap $($including both within and outside of Max$-$Heapify calls$).$ For each update, draw the corresponding binary tree representation along with the updated array. Problem $2.(10$ points$)$ We$$ve discussed four operations on a max$-$priority queue: Max$-$Heap$-$Maximum$($A$),$ Max$-$Heap$-$Extract$-$Max$($A$),$ Max$-$Heap$-$Increase$-$Key$($A$,$ i$,$ k$),$ and Max$-$Heap$-$Insert$($A$,$ k$).$ Consider the following operation Max$-$Heap$-$Delete$($A$,$ i$)$ which will remove the element at index i from a max heap. Max$-$Heap$-$Delete$($A$,$ i$)1$ if i $>$ A$.$heapsize $2$ error $$index out of range$3$ Max$-$Heap$-$Increase$-$Key$($A$,$ i$,$ $)4$ Max$-$Heap$-$Extract$-$Max$($A$)$