
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 one eigenvector for the eigenvalue $-1$ as $\begin{bmatrix} 1 \\ -3 \end{bmatrix}$, and one eigenvector for the eigenvalue $3$ as $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$,
(1) Find the matrix $M$;
(2) Find the equation of the new line $l'$ obtained from the line $l: y=2x-1$ under the transformation by $M$;
(3) Given the vector $\overrightarrow {\beta}= \begin{bmatrix} 4 \\ 0 \end{bmatrix}$, compute $M^{5}\cdot \overrightarrow {\beta}$.

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