Find the unique solution to the system
\mathbf{x}^{\prime}=\left(\begin{array}{l}x_{1} \\x_{2}\end{array}\right)^{\prime}=\left(\begin{array}{ll}4 & 2 \\3 & 3\end{array}\right)\left(\begin{array}{l}x_{1} \\x_{2}\end{array}\right)+\left(\begin{array}{l}e^{t} \\e^{2 t}\end{array}\right)=A \mathbf{x}+\mathbf{f}(t), \quad \mathbf{x}(0)=\left(\begin{array}{l}1 \\2\end{array}\right).