Solve the following system of linear equations by cross multiplication method
$2(ax - by) + a+4b=0, 2(bx+ay) + b-4a=0$

Let $x_k$ be real numbers such that $x_k\ge k^4+k^2+1$ for $1\le k\le 2018.$ Denote $N=\sum _{k=1}^{2018}k.$ Consider the following inequalities.
I. ${\left(\sum _{k=1}^{2018}k{x}_k\right)}^2\le N\left(\sum _{k=1}^{2018}k{x}_k^2\right)$

II. ${\left(\sum _{k=1}^{2018}k{x}_k\right)}^2\le N\left(\sum _{k=1}^{2018}{k}^2{x}_k^2\right)$
Then,

Consider the following statements
I. $\underset{n\to \infty }{\lim }\frac{2^n+(-2{)}^n}{2^n}$ does not exist
II. $\underset{n\to \infty }{\lim }\frac{3^n+(-3{)}^n}{4^n}$ does not exist
Then

The number of values of $k$, for which the system of equations :

$\left(k+1\right)x+8y=4k$

$kx+\left(k+3\right)y=3k-1$