RS-SMO 2010 Problem 6

Let $a_0$ and $a_n$ be different divisors of a natural number $m$ , and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies $$a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n$$ If $gcd(a_0,a_1,\ldots, a_n) = 1$ , show that there exists a term of the sequence that is smaller than $\sqrt{m}$ .

Proposed by Dusan Djukic