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:

\[ \begin{align} F1 &= 1\\ F2 &= 1\\ F3 &= 2\\ F4 &= 3\\ F5 &= 5\\ F6 &= 8\\ F7 &= 13\\ F8 &= 21\\ F9 &= 34\\ F10 &= 55\\ F11 &= 89\\ F12 &= 144 \end{align} \]

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?