$(5$ points$)$ A relative has left you $$275,000$ and you decide to invest the entire amount in two different products so that you have a larger guaranteed income of $$1500$ per month for $20$ years, starting one month from today, and then a lesser amount of X for the rest of your life. Product $1$ is a perpetuity with monthly payments at an annual effective interest rate of $5\mathrm{.}2\%$ and Product $2$ is a $20-$year annuity with monthly payments at at an annual effective interest rate of $5\mathrm{.}2\%$

i$12=6\%$

$.$ Both products make their first payment one month from the day of purchase. Find X$.$

$$547\mathrm{.}99$

$$679\mathrm{.}12$

$$885\mathrm{.}49$

$$996\mathrm{.}47$

$$1102\mathrm{.}68$