Paint the cells of a 5x5 grid with colors, such that every possible [tetromino](https://en.wikipedia.org/wiki/Tetromino) found in the grid uses 4 different colors. What is the smallest value of possible in such a coloring?

[Here](https://puzzling.stackexchange.com/questions/106814/4x4-grid-with-no-trominoes-containing-repeating-colors) is a similar question for trominoes in a 4x4 grid.