2001 AMC 12 Problem 25

Consider sequences of positive real numbers of the form x, 2000, y, \ldots 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 \qquad \text{(B) }2 \qquad \text{(C) }3 \qquad \text{(D) }4 \qquad \text{(E) more than }4

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.