KöMaL 2019 K638

Fibonacci-type sequences are defined as sequences in which, from the third term onwards, each term is the sum of the preceding two terms. For example, the sequence $1, 1, 2, 3, 5, 8, \dots$ starting with 1, 1 (the Fibonacci sequence itself), and the sequence $1, 3, 4, 7, 11, 18, 29, 47, \dots$ starting with 1, 3 are both Fibonacci-type sequences. Find the Fibonacci-type sequence that contains only positive integers, contains 2010 as a term, and has the largest possible number of terms before 2010.