, 2026

There are 1000 lily pads on a pond arranged in a circle and labelled $1, 2, \ldots, 1000$ in order. (The first four lily pads may also be described using the labels 1001, 1002, 1003 and 1004, respectively.) The first $n$ lily pads are each occupied by a frog with the remaining lily pads not occupied.

Each minute, exactly one of the frogs makes a move. Suppose the frog is on lily pad $k$. That frog may either:

(i) Swim to lily pad $k + 4$ or $k - 4$, provided that it is not occupied; or

(ii) Jump to lily pad $k + 3$ or $k - 3$ provided that this lily pad is not occupied and the two lily pads jumped over are both occupied. When this happens, the two frogs that were jumped over dive into the pond and don't participate in any further moves.

For which values of $n$ is it possible, by a sequence of moves, to end with exactly one frog remaining on the lily pads?