In Miguel's beach house, there is a magic room: whoever solves the room's riddle is granted a wish. Noticing that his friend Walfrido was sad, Thawan decided to enter the room to cheer him up. His goal is to ask for a generous portion of grapes with margarine, Walfrido's favorite snack!

However, the room will only grant the wish if Thawan can solve the following problem:

Consider a regular polygon with $$$N$$$ vertices, numbered from $$$0$$$ to $$$N - 1$$$ in a clockwise direction. Thawan starts his journey at vertex $$$0$$$. In each move, he makes a jump of size $$$S$$$, advancing exactly $$$S$$$ vertices clockwise. That is, if he is currently at vertex $$$x$$$, the jump will take him to vertex $$$(x + S) \bmod N$$$.

Knowing that Thawan can continue jumping indefinitely, determine the maximum number of distinct vertices he will be able to visit along his path. Help Thawan secure Walfrido's snack!