In triangle $$ABC,\angle{B} = 90^{\circ},AB =1$$ and $$BC = 2$$ On the side $$𝐵𝐶$$ there are two points $$𝐷$$ and $$𝐸$$ such that $$𝐸$$ lies between $$𝐶$$ and $$𝐷$$ and $$𝐷𝐸𝐹𝐺$$ is a square, where $$F$$ lies on $$𝐴𝐶$$ and $$𝐺$$ lies on the circle through $$B$$ with centre $$𝐴$$. If the area of $$𝐷𝐸𝐹𝐺$$ is $$\frac{m}{n}$$ where $$𝑚$$ and $$𝑛$$ are positive integers with $$gcd(𝑚, 𝑛) = 1$$, what is the value of $$𝑚 + 𝑛$$?