Calculating the number of rounds and dice in a game where dice are added every time a certain number is rolled

Question

I have $n$ dice with $s$ sides. With these, I play the following game:

1. Roll the unfrozen dice. All initial dice are unfrozen.
2. Freeze every die with a value below $s$
3. For every unfrozen die (i.e. each die with a value of $s$), add a new die to the game
4. Repeat steps 1-3 until all the dice are frozen.

How many rounds must be played before the game is over?

How many dice will there be at the end of the game?