Alice is $50$ years old and owns a $20-$year term endowment insurance policy that pays $$500,000.$ At age $70,$ she has the option to either claim the $$500,000$ in whole or convert it into an annuity that pays $$P$ annually at the beginning of each year starting on her $70$th birthday. Assuming that mortality follows the LTAM standard and ultimate life table and $i=5\%.$ If the APVs of the endowment insurance and annuity are equal, which value is closest to $$P?$

$$ $$32,760$

$$11,408$

$$16,174$

$$43,233$

$$46,444$