How would adding an L$3$ cache with an access time of $150$ns and a hit rate of $70\%$ affect the average access time if the L$1$ and L$2$ caches have hit rates of $90\%$ and $80\%,$ respectively $($YOU CAN USE access time $10$ns for L$1$ and $30$ns for L$2)$

HINT$-$Please think of the avg access time for a three$-$level cache system as an extension of the two$-$level system formula. You can try to calculate the avg access time for L$2$ and L$3$ first and then use that with L$1$ to calculate the total avg. You don't need to consider main memory in this case.$$

Please make sure that you explain how to calculate the average access time for a $3$ level cache system. Not just giving a formula but explain how do you come up with the formula