Consider the following page reference string, for a memory with four frames. All frames are initially empty.

$\backslash [$

$1,2,3,5,2,4,1,2,3$

$\backslash ]$

Show the result of applying $\backslash (\backslash$operatorname$\{$LRU$\}\backslash )$ and Optimal page$-$replacement algorithms to the reference string $1,2,3,5,2,4,1,2,3$

I. LRU $($Least Recently Used$)$

How many page faults would occur for the LRU page$-$replacement algorithm? $=$

II$.$ Optimal page replacement algorithm

How many page faults would occur for the Optimal page$-$replacement algorithm? $=$

III. Assume that the LRU replacement algorithm is implemented with a stack.

$1.$ Which page is at the top of the stack before reference to page $\backslash (4?=\backslash )$

$2.$ Which page is at the bottom of the stack after the reference to page $4?\backslash (=\backslash )$