Find the smallest integer k for which the conditions (1) \(a_1, a_2, a_3, \ldots\) is a nondecreasing sequence of positive integers (2) \(a_n=a_{n-1}+a_{n-2}\) for all \(n>2\) (3) \(a_9=k\) are satisfied by more than one sequence.

(第二十届AIME1 2002 第8题)