1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:

$$ F_n = F_{n−1} + F_{n−2}, where\ F_1 = 1\ and\ F_2 = 1. $$

Hence the first 12 terms will be:

$$ F_1 = 1\ F_2 = 1\ F_3 = 2\ F_4 = 3\ F_5 = 5\ F_6 = 8\ F_7 = 13\ F_8 = 21\ F_9 = 34\ F_{10} = 55\ F_{11} = 89\ F_{12} = 144 $$

The 12th term, $F_{12}$, is the first term to contain three digits.

What is the index of the first term in the Fibonacci sequence to contain 1000 digits?

Source: https://projecteuler.net/problem=25