The Coordinate Plane · Superchallenge problem

Point $P$ starts at $(2, 3)$. It is flipped (reflected) over the vertical line through $x = 5$, then over the horizontal line through $y = 8$, then over the vertical line again, then over the horizontal line again, and so on, taking turns. Where is $P$ after $2026$ flips? Write the point as (x, y).