A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On them, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). Prove that if we color the triangles black and white such that no two adjacent triangles have the same color, there will be as many rooks on the black triangles as on the white ones.