Let the curve $y=f\left(x\right)$ passes through $(4,-2)$ satisfy the differential equation, $y\left(x+y^3\right)dx=x\left(y^3-x\right)dy \&$ let $y=g\left(x\right)={\int }_{1/8}^{{\sin }^{2}x}{\sin }^{-1}\sqrt{t}dt+{\int }_{1/8}^{{\cos }^{2}x}{\cos }^{-1}\sqrt{t}dt,$ $0\le x\le \frac{\pi }{2}.$ If the area of the region bounded by curves $y=f\left(x\right), y=g\left(x\right)$ and $x=0$ is $\frac{1}{8}{\left(\frac{3\pi }{a}\right)}^4$, where $a\in N$, then $a$ is equal to