AMC 12 2001 Problem 25

Consider sequences of positive real numbers of the form $x, 2000, y, \dots$ in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of $x$ does the term $2001$ appear somewhere in the sequence?

$\text{(A) }1 \text{(B) }2 \text{(C) }3 \text{(D) }4 \text{(E) more than }4$