I would like to solve the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}$, with
$$\mathbf{A}=\mathbf{T}+\mathbf{C}$$
where $\mathbf{T}$ is a symmetric tridiagonal matrix and $\mathbf{C}$ is a corner-only matrix:
$$\mathbf{C}=\begin{pmatrix} 0& 0 & \cdots & 0 & c\\ 0 & \ddots & & & 0\\ \vdots & & \ddots & & \vdots\\ 0 & & & \ddots & 0\\ c & 0 & \cdots & 0& 0 \end{pmatrix}$$
What are some efficient algorithms for solving this system of linear equations (i.e. solving for $\mathbf{x}$)?