Grandmaster in Algebra!!!

$\text{ Non-zero real numbers a , b , c satisfy the equality }$

$\left( ab+bc+ca \right) ^{3}=abc \left( a+b+c \right) ^{3}.$

$\text{Let k denotes the expression }\frac{ \left( a-b+c \right) }{ \left( a+b+c \right) }\text{~ .}$

$\text{If the maximum and minimum values of k are x and y respectively,}$

$\text{Find the value of :}$

$\int _{0}^{xy}\frac{dt}{4^{t}+2}$