Let $f$ be a function from the non-negative integers to the non-negative integers such that $f(nm) = n f(m) + m f(n)$, with the following conditions: $f(10) = 19$, $f(12) = 52$, and $f(15) = 26$. What is the value of $f(8)$?

Let's think step by step and output the final answer within \boxed{}.
user