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:

$$ \begin{aligned} F_{1} &= 1\ F_{2} &= 1\ F_{3} &= 2\ F_{4} &= 3\ F_{5} &= 5\ F_{6} &= 8\ F_{7} &= 13\ F_{8} &= 21\ F_{9} &= 34\ F_{10} &= 55\ F_{11} &= 89\ F_{12} &= 144\ \end{aligned} $$

The $12$th 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?