XAT 2025 Permutations and Combinations — Previous Year Question

In a computer game, each move requires pressing a button. When the button is pressed for the first time, the computer randomly chooses a cell from a $4 \times 4$ grid of sixteen cells and puts an "X" mark on that cell. When the button is pressed subsequently, the computer randomly chooses a cell from the remaining unmarked cells and puts an "X" mark on that cell. This goes on till the end of the game. The game ends when either all the cells in any one row, or all the cells in any one column, are marked with "X". What is the maximum possible number of times a player has to press the button to finish the game?