
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For  $n \geq 3$  find the eigenvalues (with their multiplicities) of the  $n \times n$  matrix  $$ \begin{bmatrix}
1 & 0 & 1 & 0 & 0 & 0 & \dots & \dots & 0 & 0 
0 & 2 & 0 & 1 & 0 & 0 & \dots & \dots & 0 & 0 
1 & 0 & 2 & 0 & 1 & 0 & \dots & \dots & 0 & 0 
0 & 1 & 0 & 2 & 0 & 1 & \dots & \dots & 0 & 0 
0 & 0 & 1 & 0 & 2 & 0 & \dots & \dots & 0 & 0 
0 & 0 & 0 & 1 & 0 & 2 & \dots & \dots & 0 & 0 
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots &  & \vdots & \vdots 
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots &  & \ddots & \vdots & \vdots 
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 2 & 0 
0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 1
\end{bmatrix} $$  

Present the answer in LaTex format: \boxed{Your answer}
user