A computer has three$-$level CPU caches. The latency and miss rate of the main memory and each level cache are as follows:

main memory $=100$ ns$,$

L$1=1$ ns with $10\%$ miss rate,

L$2=5$ ns with $1\%$ miss rate,

L$3=20$ ns with $0\mathrm{.}1\%$ miss rate.

$($a$)$ What is the Average Memory Access Time $($AMAT$)$ if only L$1$ cache is used.

$($b$)$ What is the Average Memory Access Time $($AMAT$)$ if both L$1$ and L$2$ caches are used.

$($c$)$ What is the Average Memory Access Time $($AMAT$)$ if all three$-$layer caches are used.