There are $100$ stones in a pile. A partition of the heap in $k $ piles is called special if it meets that the number of stones in each pile is different and also for any partition of any of the piles into two new piles it turns out that between the $k + 1$ piles there are two that have the same number of stones (each pile contains at least one stone). a) Find the maximum number $k$, such that there is a special partition of the $100$ stones into $k $ piles. b) Find the minimum number $k $, such that there is a special partition of the $100$ stones in $k $ piles.