You are $25$ years old and want to retire at age $65.$ You have already saved $$45000$ in your retirement savings account. Each month, starting one month from now, you will deposit an equal amount into this same savings account which pays $6\mathrm{.}25\%$ interest annually, compounded monthly. The last deposit will be on your $65$th birthday. On your $65$th birthday you will switch the accumulated savings into a bank account that pays $4\mathrm{.}75\%$ interest annually, compounded monthly. You will withdraw your monthly income of $$10500$ at the end that month $($one month after your $65$th birthday$)$ and each subsequent month until your $95$ birthday. On that birthday you want to give $$875000$ to your children. How much do you have to save each month to make this retirement plan happen?