
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 a matrix $M = \begin{pmatrix} 4 & -3 \\ 2 & -1 \end{pmatrix}$, and vector $\vec{\alpha} = \begin{pmatrix} 7 \\ 5 \end{pmatrix}$,
(Ⅰ) Find the eigenvalues of matrix $M$ and an eigenvector corresponding to each eigenvalue;  
(Ⅱ) Compute $M^3 \vec{\alpha}$.

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