Buffy and Willow are facing an evil demon named Stooge, living inside Willows computer. In an effort to slow the Scooby Gangs computing power to a crawl, the demon has replaced Willows hand-designed super- fast sorting routine with the following recursive sorting algorithm, known as StoogeSort. For simplicity, we think of Stoogesort as running on a list of distinct numbers. StoogeSort runs in three phases. In the first phase, the first 2/3 of the list is (recursively) sorted. In the second phase, the final 2/3 of the list is (recursively) sorted. Finally, in the third phase, the first 2/3 of the list is (recursively) sorted again.

Willow notices some sluggishness in her system, but doesnt notice any errors from the sorting routine. This is because StoogeSort correctly sorts. For the first part of your problem, prove rigorously by induction that StoogeSort correctly sorts. (Note: in your proof you should also explain clearly and carefully what the algorithm should do and why it works even if the number of items to be sorted is \textbf{not} divisible by 3. You may assume all numbers to be sorted are distinct.) But StoogeSort can be slow. Derive a recurrence describing its running time, and use the recurrence to bound the asymptotic running time of Stoogesort.