A Fibonacci-like sequence satisfies $a_n = a_{n-1} + a_{n-2}$ with $a_0 = 0$ and $a_1 = 1$. The characteristic equation has roots $\phi = \frac{1+\sqrt{5}}{2}$ and $\psi = \frac{1-\sqrt{5}}{2}$. What is $a_5$?