$4.$ Amortized Analysis $($a$)(10$ points$)$ Consider a $($distributed$)$ data structure that uses computational puzzles to control membership. There is only one operation provided: JOIN. This allows nodes to join the system by solving a puzzle with a computational cost of $1.$ In addition, there are periodic purges where all nodes in the system must again solve a puzzle with a computational cost of $1.$ A purge occurs when the number of JOINs since the last purge is $1/3$ the size of the system after the last purge. What is the amortized cost of the JOIN operation? Show your work. Hint: Use the accounting method, and make sure you tax JOIN enough to pay for all puzzle costs $($at entrance and during all purges$).$