The famulus of a mathematical meta magician puts infinitely many cards
into a top hat.
Each of these cards carries a non-negative integer.
The top hat contains
- exactly $n$ cards showing the integer $0$,
- exactly $n$ cards showing the integer $1$,
- exactly $n$ cards showing the integer $2$,
- exactly $n$ cards showing the integer $3$,
- exactly $n$ cards showing the integer $4$,
- and so on.
Then the mathematical meta magician starts removing cards from the top hat.
- In the $1$st step, the magician takes ten cards with total value $1$.
- in the $2$nd step, the magician takes ten cards with total value $2$,
- in the $3$rd step, the magician takes ten cards with total value $3$,
- in the $4$th step, the magician takes ten cards with total value $4$,
- in the $5$th step, the magician takes ten cards with total value $5$,
- and so on,
- in the $k$th step, the magician takes ten cards with total value $k$,
- and so on.
Question: What is the smallest possible integer $n$, for which the mathematical meta magician can make an infinite number of such steps (without ever getting stuck)?