Middle European Mathematical Olympiad 2025 Problem I-2

On an infinite square grid, on which some unit squares are coloured red, a ruby rook is a piece which, in one move, can travel any number of squares in one direction parallel to one of the grid lines (either vertically or horizontally), while remaining on red squares at all times throughout the move.

Starting with an uncoloured infinite square grid, Alice performs the following procedure: First, she colours at most 2025 of the unit squares red. Afterwards, she places some ruby rooks on distinct red unit squares, such that the following two rules are satisfied:

• No ruby rook can reach another ruby rook in one move.
• Every ruby rook can reach every other ruby rook in two moves.

Find the maximum possible number of ruby rooks that Alice can place during this procedure.