Consider a computer uses $48-$bit byte$-$addressable memory address, and it has a memory of $4$ GB and a one$-$level cache of $2$ KB $($not counting the space for tags nor dirty flags$).$ Suppose a cache block is composed of $32$ words and a word consists of $4$ B $.$

Note: $1$ G denotes the number $\backslash (2^\{30\}\backslash ),$ and $1$ K denotes the number $\backslash (2^\{10\}\backslash ).$

$($a$)$ Please derive the number of bits required by different fields $($i$.$e$.,$ Tag, Block, Set, Word, and Byte$)$ when the following types of cache mapping functions are used respectively: $1)$ direct mapping, $2)$ associative mapping, and $3)$ two$-$way set$-$associative mapping.

$($b$)$ Suppose that the cache was empty initially, and the processor has fetched $576$ consecutive words starting at memory address $0.$ Assume that now the processor wants to consecutively fetch these $576$ words one more time $($again$,$ starting from memory address $0),$ please derive the cache hit rate$($s$)$ for the second round of $576$ consecutive word accesses when the following cache mapping functions and replacement algorithms are used respectively: $1)$ direct mapping, $2)$ associative mapping with LRU replacement, and $3)$ two$-$way set$-$associative mapping with LRU replacement.