3. Given a complex number $z$ satisfies $\arg (z+a+a \mathrm{i})=\frac{\pi}{4}$, and $\arg (z-a-a \mathrm{i})=\frac{5 \pi}{4}\left(a \in \mathbf{R}^{+}\right)$, find the range of $\arg (z-b+b \mathrm{i})(a>b>0)$.
3. The complex number representing point $P$ on line segment $AB$ simultaneously satisfies the equations $\arg (z+a+a \mathrm{i})=\frac{\pi}{4}, \arg (z-a-a \mathrm{i})=\frac{5 \pi}{4}$, where $A(a, a), B(-a,-a) \cdot \arg (z-b+b \mathrm{i})$ can be considered as the angle from the positive direction of the $x$-axis to the ray $CP$, where $C(b,-b)$. The slopes of $AC$ and $BC$ are $k_{A C}=\frac{a+b}{a-b}, k_{B C}=\frac{a-b}{a+b}$, so the range of $\arg (z-b+b \mathrm{i})$ is $\left[\arctan \frac{a+b}{a-b}, \pi+\right.$ $\left.\arctan \frac{a-b}{a+b}\right]$.