Wythoff's queen race

A queen sits away from the corner and may move only west, south, or south-west, always toward a1. Players alternate, and whoever lands on a1 wins. Because you can never move away from the corner, the game must end, and every position is either winning or losing for the player to move.

The losing positions, the cold squares, are the ones you want to hand your opponent. They are not random: their coordinates follow Wythoff pairs built from the golden ratio, and the first few, counting a1 as $(0, 0)$, are $(1, 2)$ and $(2, 1)$. Find a move onto a cold square and you keep control.