$1.$ Construct a $2^{5-2}$ fractional factorial model using the optimal generators $($showing your design matrix and selected observed values$).$ Comment on the alias structure and any concerns about interpreting the results.

$2.$Calculate the Estimated Effects for each of the five main factors $($not for the interaction effects$)$ and identify those with the strongest effects.

$3.$ Calculate the Sum of Squares values and complete the ANOVA table. Comment on all significant effects and identify important alias relationships.

$4.$ Consider that you have a three$-$level nuisance factor to add to this design. Discuss approaches that you could use to implement the blocks in this model $($your response should have at least two approaches$).$