# 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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