Consider the function $f(x) = \log_{a}(x^2 - bx + c)$. If the vertical asymptotes are at $x = 1$ and $x = 5$, and the function satisfies $f(3) = 0$, what is the value of $a + b + c$ assuming $a > 1$ and the base is such that $f(6) = 1$?