Another system of equations

Algebra Level 5

\(\left\{ \begin{gathered} 2{x^2} + xy - {y^2} = 1 \\ {x^2} + xy + {y^2} = m \\ \end{gathered} \right.\)

For all \(m \ge \displaystyle\frac{a+b\sqrt{c}}{d} \), the equation has at least one real root pair \(x,y\)

If \(a,b,c,d \in \mathbb{Z}\), \(c\) is square-free and \(\gcd(a,b,d) = 1\), find \(a+b+c+d\)

Hint: It is recommended that the problem be solved graphically.

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