TxO Math Bowl 2024 - Team Contest - Problem 21

In a room, there are 5 light bulbs turned off, each paired up with a button. At the end of every minute, Alice presses a button and turns the corresponding light bulb on. The button can be pressed even if the light is currently on. If n is the total number of times a button was pressed, the corresponding light bulb will inexplicably switch back off n^2 minutes after the latest button press. Following optimal strategy, what is the minimum amount of time until Alice is able to turn on all lights, even if just for a moment?