KöMaL 2021 A794

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $10^6 + 1$ are there?