Joel invested $$800$ at the end of every month into an investment fund that was earning interest at $4\mathrm{.}50\%$ compounded monthly. She stopped making regular deposits at the end of $10$ years when the interest rate changed to $3\mathrm{.}50\%$ compounded quarterly. However, she let the money grow in this investment fund for the next $3$ years. Calculate the accumulated balance in her investment fund at the end of $13$ years.

Question $3$ options:

$$169249\mathrm{.}67$

$$134,288\mathrm{.}49$

$$120,958\mathrm{.}46$

$$165,517\mathrm{.}52$