Solve Matrix

$$\left. \begin{array} { l } { y = \sin ^ { 3 } \frac { x } { 3 } } \\ { z = \int _ { 0 } ^ { 2 \pi } \sqrt { ( \frac { d y } { d x } ) ^ { 2 } + y ^ { 2 } } d x } \end{array} \right.$$

Solve for y, x, z

$$\left\{\begin{matrix}\\x=3\left(-\arcsin(\frac{2^{\frac{2}{3}}\sqrt[3]{\frac{z}{\pi }}}{2})+2\pi n_{1}+\pi \right)\text{, }n_{1}\in \mathrm{Z}\text{, }y=\left(\sin(\frac{x}{3})\right)^{3}\text{, }z\in \begin{bmatrix}0,2\pi \end{bmatrix}\text{; }x=3\left(\arcsin(\frac{2^{\frac{2}{3}}\sqrt[3]{\frac{z}{\pi }}}{2})+\pi n_{4}\right)\text{, }n_{4}\in \mathrm{Z}\text{, }y=\left(\sin(\frac{x}{3})\right)^{3}\text{, }z\in \begin{bmatrix}0,2\pi \end{bmatrix}\text{, }&\text{unconditionally}\\x=3\left(-\arcsin(\frac{2^{\frac{2}{3}}\sqrt[3]{\frac{z}{\pi }}}{2})+2\pi n_{2}\right)\text{, }n_{2}\in \mathrm{Z}\text{, }\exists n_{3}\in \mathrm{Z}\text{ : }\left(z\geq -2\pi \left(\sin(2\pi \left(n_{3}+1\right))\right)^{3}\text{ and }n_{2}=n_{3}+1\right)\text{, }y=\left(\sin(\frac{x}{3})\right)^{3}\text{, }z\in \begin{bmatrix}0,2\pi \end{bmatrix}\text{, }&\exists n_{3}\in \mathrm{Z}\text{ : }z\geq -2\pi \left(\sin(2\pi \left(n_{3}+1\right))\right)^{3}\end{matrix}\right.$$