Consider the matrix $$M=\begin{bmatrix}2&-1\\1&0\end{bmatrix}.$$

Let $$p,q,r,s,a,b,c$$ and $$d$$ be integers such that $$M^{26}=\begin{bmatrix}p&q\\r&s\end{bmatrix}$$ and $$\displaystyle\sum_{k=1}^{26}M^k=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$

Then which of the following statements is (are) TRUE?

A

There exists a $$2\times 2$$ invertible matrix $$N$$ with real entries such that $$MN=N\begin{bmatrix}1&1\\0&1\end{bmatrix}$$

B

The value of $$a$$ is $$378$$

C

For any two given integers $$m$$ and $$n$$, there exist unique integers $$x$$ and $$y$$ such that $$px+qy=m$$ and $$rx+sy=n$$

D

For each positive real number $$t$$, the system of linear equations $$(a+t)x+by=1,\;cx+(d+t)y=-1$$ has a unique solution