Expected Product

Chef is playing a tricky game with a sequence $a_1, a_2, \ldots, a_N$. Initially, his score is $S = 1$. In the game, he performs the following process exactly $K$ times (note that the sequence $a$ never changes): - Pick an element $X$ from the sequence $a$ uniformly randomly. - Change the score $S$ to $(S \cdot X) \% N$. Find the expected value of Chef's score $S$ at the end of the game. It can b