$3.(35$ pts$)$ In a hypothetical paged virtual memory system, in which the following parameters are defined: $(1)6-$bit virtual word address, $(2)8-$word page size, $(3)32$ words of main memory, $(4)1-$byte page table entry $($PTE$),(5)$ fully associative main memory placement policy, $(6)8$ words of cache, $(7)4-$word block size, $(8)$ direct mapped cache placement policy, answer the following questions:

$($a$)$ what is the size of the largest user program this system can handle? $($i$.$e$.$ what is the usable disk space for programs?$)$

$($b$)$ what is the largest size of storage space that a single page table can occupy?

$($c$)$ when executing a program, the CPU generates a sequence of virtual word addresses as in the following:

$0$x$05,0$x$1$b$,0$x$2$b$,0$x$18,0$x$26,0$x$04,0$x$24,0$x$1$e

and its corresponding page table has the following contents: $($where each row corresponds to a PTE, and VPN$,$ V$,$ xxx and PPN each stands for virtual page number, valid bit, irrelevant bits, and physical page number, respectively$)$

and the cache has its current snapshot shown as in the following table: $($where SA stands for Set Address for cache placement policy$)$

disregarding the accesses to page table, how many page faults and cache misses will the system encounter through this sequence of accesses, assuming that no other programs are being run in the system $($even a page fault happens$)$ and the whole memory is available to this program $($also disregard the memory requirement for page table$)?$ Show at which address access$($es$)$ the page fault$($s$)$ and cache miss$($s$)$ are encountered.