
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 second-order matrix $M$ with eigenvalue $\lambda_1=4$ and an eigenvector belonging to the eigenvalue 4, $\overrightarrow{e_1} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, and eigenvalue $\lambda_2=-1$ and an eigenvector belonging to the eigenvalue -1, $\overrightarrow{e_2} = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$, $\overrightarrow{\alpha} = \begin{pmatrix} -1 \\ 1 \end{pmatrix}$  
(Ⅰ) Find the matrix $M$;  
(Ⅱ) Calculate $M^5 \overrightarrow{\alpha}$.

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