You are given $$20,000$ to invest for the next $40$ years.

$($a$)$ Plan $1$ promises $7$ percent per year for the first $20$ years and $11$ percent per year for the last

$20$ years. Plan $2$ promises $11$ percent per year for the first $20$ years and $7$ percent per year for

the last $20$ years. Which plan is better?

$($b$)$ Plan $3$ promises $7$ percent per year for the first $30$ years and $11$ percent per year for the last

$10$ years. Plan $4$ promises $11$ percent per year for the first $30$ years and $7$ percent per year for

the last $10$ years. Among Plans $3$ and $4,$ which plan is better?