
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
Given the matrix $M= \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}$, and $a= \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.

(1) Find the inverse matrix $M^{-1}$ of matrix $M$;  
(2) Find the eigenvalues and eigenvectors of matrix $M$;  
(3) Calculate $M^{20}a$.

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