Points, Lines & Planes · Superchallenge problem

Let $A = (3, 0)$, $B = (5, 4)$, $C = (0, 7)$, and $P_0 = (1, 2)$. Define points $P_1, P_2, P_3, \ldots$ so that $A$ is the midpoint of $\overline{P_0P_1}$, $B$ is the midpoint of $\overline{P_1P_2}$, $C$ is the midpoint of $\overline{P_2P_3}$, $A$ is the midpoint of $\overline{P_3P_4}$, and so on, the midpoints cycling through $A, B, C$. Find the distance $P_0P_{2026}$. Give your answer in simplest radical form.