Let $a$ and $b$ be real numbers such that \[\begin{cases} a^6 - 3a^2b^4 = 3 \\ b^6 - 3a^4b^2 = 3\sqrt{2}.\end{cases}\] What is the value of $a^4 + b^4$?

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