A particle starts from a point $z = 1 + i$, where $i = \sqrt{-1}$. It moves horizontally away from origin by $2$ units and then vertically away from origin by $3$ units to reach a point $z_1$. From $z_1$ the particle moves $\sqrt{5}$ units in the direction of $2\hat{i} + \hat{j}$ and then it moves through an angle of $\csc^{-1}(\sqrt{2})$ in anticlockwise direction around a circle with centre at origin to reach a point $z_2$. The $\arg(z_2)$ is given by