$($C$)$ Harvey sells a high$-$end espresso machine, an expensive and bulky item. For this reason, Harvey attaches a very high holding cost of $$70$ per year to each machine. Harvey sells about $500$ of these yearly, and estimates the variance of yearly demand to be $144.$ Annual demand follows a normal distribution. Since customers are willing to wait for the machine when he is out of stock, the shortage penalty is low. He estimates the shortage penalty to be $$35$ per unit. Order lead time is three months. The fixed ordering cost is $$180.$ Determine the following:

$($a$)$ The lot size and the reorder point, using the iteration approach. Start with LaTeX: Q$\_0=$EOQ. What are the values of LaTeX: Q$\_1$ and LaTeX: R$\_1($round to the nearest integers$)?$

$($b$)$ What is the safety stock?

$($c$)$ What is the resulting Type $1$ service level?

$($d$)$ What is the resulting Type $2$ service level?