$3.$ Shelley wants to buy an apartment in five years. She estimates it will cost $$250000$ and she needs a $10\%$ deposit. She makes monthly deposits to a bank earning $12\%$ p$.$a$.,$ compounded quarterly until she is ready to buy.

a$)$ How much deposit does Shelley need for the apartment?

b$)$ How much should she set aside each month to reach her deposit goal?