A Typical Area

Two curves C $\equiv { \left( f\left( x \right) \right) }^{ \frac { 1 }{ 3 } }+{ \left( f\left( y \right) \right) }^{ \frac { 2 }{ 3 } }=0$ and

S $\equiv { \left( f\left( x \right) \right) }^{ \frac { 2 }{ 3 } }+{ \left( f\left( y \right) \right) }^{ \frac { 2 }{ 3 } }=12$ ,

satisfying the relation ${ \left( { x }-{ y } \right) }{ f\left( x+y \right) }-{ \left( { x }+{ y } \right) }{ f\left( x-y \right) }={ { 4 }{ x }{ y } }{ \left( { x }^{ 2 }-{ y }^{ 2 } \right) }$ . where f(1)=1

Let A represent area between the curves C and S and A can be expressed as ${ p }{ \pi }+\sqrt { q }$
Then find the value of p+q+ $\left\lfloor A \right\rfloor$ .