$(20$ Points$)$ In the coagulation reaction, commercial aluminum sulfate $($alum$)$ reacts with natural alkalinity or can be reacted with lime or soda ash if the water is deficient in alkalinity. Based on the reactions below, calculate the reduced $m\frac{g}{1}$ alkalinity as CaCO $3,$ lime as Cao, and soda ash as Na $2$ CO $3$ that react with $1m\frac{g}{l}$ alum.