We have the following accumulation function:

$a(t)=\{\begin{array}{ccc}a{t}^2+bt+c,& \text{f}\text{o}\text{r}& 0t6,\\ (a{t}^2+bt+c)(1+0\mathrm{.}05(t-6)),& \text{f}\text{o}\text{r}& t>6\end{array}$

Assuming that $i_3=\frac{50}{1088}$ and $d_4=\frac{54}{1192}$

a$)$ Determine $a,b,$ and $c.$

b$)$ How much would need to be invested at $t=3$ to accumulate $$1,000$ by $t=8?$