Problem $4$: You receive $$4,000$ from your aunt when you turn $21$ and you immediately invest the money in a saving account. The account earns $12\%$ annual rate, with continuous compounding. You get your first job after $5$ years. You want to retire from work in $20$ years. If you deposit $$100$ into your account every month for the first $10$ years, and $$200$ every month for the next $10,$ how much will you have after $20$ years? Assume you continue to earn $12\%$ annual rate with continuous compounding?