Wood Watercolors is a commercial art gallery that uses a four$-$week periodic review system to reorder paintings for its showrooms. A one$-$week lead time is required to fill the order. Demand for one particular style of painting during the five$-$week replenishment period is normally distributed with a mean of $22$ units and a standard deviation of $10$ units.

$($a$)$ At a particular periodic review, $12$ units are in inventory. The curator places an order for $20$ units. What is the probability that this style of painting will have a stock$-$out before an order that is placed during the next four$-$week review period arrives? $($Round your answer to four decimal places.$)$

$($b$)$ Assume that the gallery is willing to tolerate a $2\mathrm{.}5\%$ chance of a stock$-$out associated with a replenishment decision. How many paintings should the curator have ordered in part $($a$)?($Round your answer to the nearest integer.$)$

$($c$)$What is the replenishment level for the four$-$week periodic review system? $($Round your answer to the nearest integer.$)$