Let $$a, b, c$$ be the lengths of three sides of a triangle satistying the condition $$\left(a^2+b^2\right) x^2-2 b(a+c) x+\left(b^2+c^2\right)=0$$. If the set of all possible values of $$x$$ is the interval $$(\alpha, \beta)$$, then $$12\left(\alpha^2+\beta^2\right)$$ is equal to __________.

The number of real solutions of the equation $$x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$$ is _________.

Let $$\alpha, \beta \in \mathbf{N}$$ be roots of the equation $$x^2-70 x+\lambda=0$$, where $$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbf{N}$$. If $$\lambda$$ assumes the minimum possible value, then $$\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$$ is equal to :

Let the set $$C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}$$. Then $$\sum_\limits{(x, y) \in C}(x+y)$$ is equal to _________.