A king has 100 bottles of wine and poisons $K$ of them, where $0 \leq K \leq 100$. You have a supply of rats and need to determine how many of the bottles are poisoned. You are not interested in finding the actual poisoned bottles and you only want to find the value of $K$. You can get a rat to drink a mix of wine taken from different bottles. If the mix contains any poisoned wine then the rat will die after 1 hour. What is the minimum number of rats required to conclusively identify the value of $K$ in the general case (not any specific $K$)?
I only know the obvious solution that uses 100 rats, so I am very interested in seeing if better solutions exist! Note I am leaving this puzzle open to either adaptive (rats can be reused, but longer wait time) or non-adaptive (rats cannot be reused, but just 1 hour wait time) strategies. I am interested in both types of solutions.
P.S. No rats were harmed in the making of this puzzle :)