$(20$ pts$)$ You are managing a ticket printing exchange machine. It works by accepting a ticket, with a non$-$negative integer number, $t,$ as input. It can then produce a new ticket $($destroying the input one$)$ which has a non$-$negative integer number on it equal to either $t^2$mod$400$ or $t^2+1$mod$400.$ For example, if you give it a ticket with $t=103,$ it will give you a ticket with values of $209$ or $210.$ You get to pick which of the tickets you want to take, but can only select one of them.

Assuming that you start with a ticket with $t=1,$ is it possible to obtain a ticket with $t=20$ from the machine? You may exchange as many tickets as you like, feeding your new ticket back into the machine. Model the problem using a graph, giving a precise definition of your vertices and edges, and cast the problem statement in terms of graph concepts. Then provide an algorithm which uses this graph model to solve the problem, and state the time complexity of your solution.