The Fibonacci numbers satisfy $F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}$ where $\phi = \frac{1+\sqrt{5}}{2}$ and $\psi = \frac{1-\sqrt{5}}{2}$. A related sequence $G_n$ satisfies the same recurrence $G_n = G_{n-1} + G_{n-2}$ but with initial conditions $G_0 = 1, G_1 = 1$. Compute $G_5$.