Suppose that a pension fund manager knows that she will need $$10,000,000$ in $10$ years to satisfy a liability at that time. She believes that she can earn an annual return of $8\%$ compounded semiannually for the first $6$ years and $5\%$ compounded semiannually for the last $4$ years. How much does she need to invest today $($one$-$time investment$)$ to ensure that she has $$10,000,000$ in $10$ years?