Solve the system of differential equations for $x(t), y(t), z(t)$,

$$\begin{aligned} &\dot{x}=3 z-x \\ &\dot{y}=3 x+2 y-3 z+\cos t-2 \sin t \\ &\dot{z}=3 x-z \end{aligned}$$

subject to the initial conditions $x(0)=y(0)=0, z(0)=1$.