If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$, where $a, b, c, d$ are integers, what is the value of $|a + b + c + d|$?

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