linear algebra - How to find eigenvalues of the matrix

This is a question from our end-semester exam:

How to find the eigenvalues of the given matrix:

M=

$$\begin{bmatrix} 5,1,1,1,1,1\\ 1,5,1,1,1,1\\ 1,1,5,1,1,1\\ 1,1,1,5,1,1\\ 1,1,1,1,4,0\\ 1,1,1,1,0,4\\ \end{bmatrix}$$

I know that $4$ is an eigenvalue of $M$ with multiplicity atleast $3$ since $M-4I$ has $4$ identical rows.

Is there any way to find all eigenvalues of this matrix? I could find only $3$ out of $6$.