Consider two stacks A and B manipulated using the following operations

$($n is the size of A and m the size of B$)$:

$$ P ushA$($x$)$: Push element x on stack A$.$

$$ P ushB$($x$)$: Push element x on stack B$.$

$$ M ultiP opA$($k$)$: Pop min$\{$k$,$ n$\}$ elements from A$.$

$$ M ultiP opB$($k$)$: Pop min$\{$k$,$ m$\}$ elements from B$.$

$$ T ransf er$($k$)$: Repeatedly pop an element from A and push it on B$,$ until either k elements

have been moved or A is empty.

Your task is to

$(1)$ define a potential function $\backslash$Phi $($n$,$ m$)$ and use it to prove that the five operations have amor$-$

tized running time O$(1).$

$(2)$ determine the total cost of executing n of the operations PushA, PushB, and MultiPopA,

MultiPopB, and Transfer, assuming that stack A and B begin with s$0,$ t$0$ objects and finish

with sn$,$ tn objects respectivel