Fibonacci Partitions

105 is the smallest integer which can be decomposed in 10 ways as sum of distinct Fibonacci numbers:

1 2 5 8 13 21 55

1 2 5 8 34 55

1 2 5 8 89

1 2 13 34 55

1 2 13 89

3 5 8 13 21 55

3 5 8 34 55

3 5 8 89

3 13 34 55

3 13 89

What is the smallest integer with 5000 decompositions ?

[My timing: 80 sec]