Kontsevich’s puzzle

Consider the following diagram, in which there is an infinite number of stones in a row, and only the top stone is white and the rest are black.

           ●
         ○   ○
       ○   ○   ○
     ○   ○   ○   ○
  ○   ○   ○   ○   ○
.....................

Prove that no matter how you repeat the following operations, there will remain a white stone anywhere from the top to the fourth step.

[Operation] Choose a white stone such that both the bottom left and the bottom right stones are black, change it to black, and change the bottom left and the bottom right stones to white.

Comments

This puzzle is based on the Maxim Kontsevich’s puzzle in 1981: Escape of the Clones.