AIME 2018b Problem 15

Find the number of functions $f$ from $\{0, 1, 2, 3, 4, 5, 6\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and $$|x - y| \leq |f(x) - f(y)| \leq 3|x - y|$$ for all $x$ and $y$ in $\{0, 1, 2, 3, 4, 5, 6\}$.