Let $x,y,z \in (-1;1)$ such that $$13\left( {xy + yz + zx + 1} \right) + 14\left( {x + y + z + xyz} \right) = 0.$$ Find the minimum value of the following expression $$P=x^2+y^2+z^2.$$
I guess the minimum value is $\frac{3}{4}$ and I will try to prove that prediction. From the conditions of the variables, I thought of trigonometry. However, it cannot be used due to the conditions $$13\left( {xy + yz + zx + 1} \right) + 14\left( {x + y + z + xyz} \right) = 0.$$ I also thought of the $pqr$ method, but the variables are real numbers (which can be negative), so I couldn't implement the idea. Please provide some suggestions. Thank you.