A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0,1,2,\ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?

[i]Proposed by Brandon Wang[/i]
**
   Since the recurrence relation and the initial conditions suggest that \( T_k \) converges to a constant value, we conclude that:
   \[
   T_{2018} = 2019
   \]

The final answer is \( \boxed{2019} \).