Unbounded set in $\mathbb{R}^2$

Question

Let $C>0$. We consider the set

$A(C)=\left\{(x,y)\in\mathbb{R}^2, \left(2\left|xy\right|\sqrt{x^2+y^2}\right)^{\frac{4}{3}}\le C\left(\sqrt{\left(x^2+y^2\right)^2+6x^2y^2}+\left(\left|4x\right|^{\frac{1}{3}}+\left|4y\right|^{\frac{1}{3}}+2\left|4\right|^{\frac{1}{3}}\right)^4\right)\right\}$

Please help me to prove that for all $C>0$ the set $A(C)$ is not bounded.

Thanks in advance