Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire $30$ years from now with retirement income of $$23,000$ per month for $20$ years, with the first payment received $30$ years and $1$ month from now. Second, he would like to purchase a cabin in Rivendell in $10$ years at an estimated cost of $$337,000.$ Third, after he passes on at the end of the $20$ years of withdrawals, he would like to leave an inheritance of $$700,000$ to his nephew Frodo. He can afford to save $$2,000$ per month for the next $10$ years.

If he can earn a $11$ percent EAR before he retires and a $8$ percent EAR after he retires, how much will he have to save each month in Years $11$ through $30?$