A money worth $\$3000$ is deposited in a $4-year$ savings account that pays a fixed annual interest rate $r$ (expressed as a decimal) with quarterly compounding. The amount in the account after $4$ years is given by $a\left(r\right)=3000\left(1+\frac{r}{4}\right)^{16}$. Find the value of $r$ (using Newton’s method) needed for the account to grow to $\$4000$ after $4$ years.