It is desired to accumulate $$5,000$ at the end of $12$ years. The account credits interest at an effective annual rate of $4\%$ for the first four years, $5\%$ for the next four years, $5\mathrm{.}5\%$ for the next three years, $6\%$ for the next two years, and $7\%$ for the next three years.

Using the effective interest rates, what is the amount that would have to be invested at the end of the second year?