Mr$.$ Burns, age $8080,$ wants to retire at age $100.$ Once Mr$.$ Burns$$ retires, he wants to withdraw $$$$0\mathrm{.}90\mathrm{.}9$ billion at the beginning of each year for $1010$ years from an account that will pay $2525$ percent annually. In order to fund these$$ withdrawals, Mr$.$ Burns will make $2020$ equal$$ end$-$of$-$the$-$year deposits in the same account$($i$.$e$.,$ one that will pay $2525$ percent$$ annually$).$

$($a$)$ How much money will Mr$.$ Burns need at age$100,$ and$($b$)$ how large of an annual deposit must he make to fund this retirement$$ account?

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Part $1$

a$.$ How much money will Mr$.$ Burns need when he$$ retires?

$$$enter your response here billion $($Round to three decimal$$ places.$)$

b$.$ How large of an annual deposit must he make to fund this retirement$$ account?