Is it Symmetric?

Algebra Level 5

Let $$x_{1} , x_{2} , x_{3} ,x_{4}$$ be real numbers such that

$$\frac{1}{2}\leq x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} \leq 1$$

$$A = (x_{1} - 2x_{2} + x_{3})^{2} + (x_{2} - 2x_{3} + x_{1})^{2} + (x_{2} - 2x_{1})^{2} + ( - 2x_{4} + x_{3})^{2}$$

$$A \in [ \frac{a -b\sqrt{c}}{d} , \frac{e + f\sqrt{g}}{h}]$$

Find $$a + b + ........... + h$$

Note - It contains the favourite number of Sandeep Bhardwaj and Sanjeet Raria , when A is maximum

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