The coefficient of $x^{12}$ in $(x^3 + x^4 + x^5 + x^6 + \cdots)^3$ is ________ .

We want to find the coefficient of $x^{12}$ in the expansion of:

Factor out $x^3$ from each term in the product:

Finding the coefficient of $x^{12}$ in $S$ is equivalent to finding the coefficient of $x^{12 - 9} = x^3$ in $(1 - x)^{-3}$.

Using the binomial series expansion:

For $n = 3$ and $r = 3$:

Thus, the coefficient of $x^{12}$ is 10.