In this problem, we will explore how deepening the pipeline affects performance in two ways: faster clock cycles and increased stalls due to data and control hazards. Assume that the original machine is a $5-$stage pipeline with a $1$ns clock cycle. The second machine is a $12-$stage pipeline with a $0\mathrm{.}6$ ns clock cycle. The $5-$stage pipeline experiences a stall due to a data hazard every five instructions, whereas the $12-$stage pipeline experiences three stalls every eight instructions. In addition, branches constitute $20\%$ of the instructions, and the misprediction rate for both machines is $5\%.$

a$).$ What is the speedup of the $12-$stage pipeline over the $5-$stage pipeline, taking into only data hazards?

b$).$ If the branch mispredict penalty for the first machine is $2$ cycles but the second machine is $5$ cycles, what are the CPIs of each, taking into account the stalls due to branch misprediction?