You are a new employee with the Metro Daily Planet. The Planet offers three different retirement plans. Plan $1$ starts the first day of work and puts $$$$1$ comma $400$ away in your retirement account at the end of every year for $40$ years. Plan $2$ starts after $10$ years and puts away $$$$1$ comma $900$ every year for $30$ years. Plan $3$ starts after $20$ years and puts away $ $4$ comma $900$ every year for the last $20$ years of employment. All three plans guarantee an annual growth rate of $8\%.$

a$.$Which plan should you choose if you plan to work at the Planet for $40$years$?$

b$.$ Which plan should you choose if you plan to work at the Planet for only the next $30$years$?$

c$.$ Which plan should you choose if you plan to work at the Planet for only the next $20$years$?$

d$.$ Which plan should you choose if you plan to work at the Planet for only the next $10$years$?$

e$.$ What do the answers in parts$($a$)$ through$($d$)$ imply about$$ savings?