For \(n \ge 0\), let \(F_n\) denote the \(n\)th Fibonacci number (that is, \(F_0 = 0, F_1 = 1\), and \(F_n = F_{n-1} + F_{n-2}\) for all \(n \ge 2\)). What is the greatest possible value of the greatest common divisor of two consecutive Fibonacci numbers?
