John is planning his retirement which will occur in the future. During retirement, he wishes to withdraw $$12,000$ per month from his portfolio for the first $10$ years and $$20,000$ per month for the second $10$ years. $($He plans to withdraw from this investment for a total of $20$ years.$)$ In addition, John want to leave $$1,000,000$ to his heirs at the end of the retirement period. John plans to earn $4$ percent compounded monthly during retirement.

John plans to save $$300$ per month in an asset that earns $13$ percent compounded monthly. How many years must he save to fully fund his retirement and inheritance? $($Assume monthly compounding.$)$

Beginning Amount $=0$

Savings amount $=$ $$300$ per month

Rate during Savings $($APR$)=13\mathrm{.}00\%$ annual

Monthly Compounding $($M$)=12$

Length of retirement period $=20$ years

First Withdrawal Period Length $=10$ years

Second Withdrawal Period Length $=10$ years

Rate during Retirement $=4\%$

Withdrawal amount $1$st period $=$ $$12,000$ per month

Withdrawal amount $2$nd period $=$ $$20,000$ per month

Inheritance $=$ $$1,000,000$

Monthly compounding $($M$)=12$