Graphs of the Other Trigonometric Functions · Superchallenge problem

For some $\varphi$ with $0<\varphi<\pi$, the graph of $y=3\tan(2x-\varphi)+1$ is symmetric about the point $\left(\dfrac{\pi}{3},\,1\right)$ (a half-turn about that point maps the graph onto itself). Find the sum of all possible values of $\varphi$.