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:

$$\begin{array}{rl}{F}_1& =1\\ F_2& =1\\ F_3& =2\\ F_4& =3\\ F_5& =5\\ F_6& =8\\ F_7& =13\\ F_8& =21\\ F_9& =34\\ F_{10}& =55\\ F_{11}& =89\\ F_{12}& =144\end{array}$$

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?