$20\%$ We extend the Tower of Hanoi problem to have $4$ pegs while all rules

remain $($i$.$e$.,$ every time, you can only move one disk at a time and cannot put a

larger disk over a smaller one$).$

$($a$)$ Design a recursive algorithm to solve the problem of Tower of Hanoi with

$4$ pegs and provide a recursive function to capture the time complexity of

your algorithm $($NO need to solve the recursive function to get the closed

form$).$

$($b$)$ Based on your algorithm, please provide the number of moves used by

your algorithm if $($i$)$ there are $7$ disks; or $($ii$)$ there are $15$ disks.