There is an array with n integers, but the values are hidden to us. Our goal is to partition the elements into groups based on their values — elements in the same group should have the same value, while elements in different groups have different values. The values are hidden to us, but we can probe the array in the following way: we can query a subset of these n elements, and get the number of unique integers in this subset. Design an algorithm to partition these n elements in O(n log n) queries.

Write a program that performs a simple n-body simulation, called "jumping leprechauns." this simulation involves n leprechauns, numberd 1 to n. it maintains a gold value g_i for each leprechaun i, which begins with each leprechaun starting out with a million dollars worth of gold, that is, g_i = 1000000 for each i = 1,. in addition, the simulation also maintains, for each leprachaun,i, a place on the horizon, which is represented as a double-precision floating point number, x_i. in each iteration of the simulation, the simulation processes the leprachauns in order. processing a leprachaun i during its iteration begins by computing a new place on the horizon for i, which is determined by the assignment: