Big Fibonacci

The fifth term of a Fibonacci sequence is 2004. If all the terms are positive integers, what is the largest possible first term?
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Problem



In a sequence of positive integers, every term after the first two terms is the sum of the two previous terms in the sequence.

If the fifth term is $2004$, what is the maximum possible value of the first term?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.