
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 $\lambda=8$ and a corresponding eigenvector $\overrightarrow{e_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, and the transformation corresponding to matrix $M$ maps the point $(-1, 2)$ to $(-2, 4)$,
(1) Find the matrix $M$;
(2) Find another eigenvalue of matrix $M$, and the relationship between the coordinates of the corresponding eigenvector $\overrightarrow{e_2}$;
(3) Find the equation of the line $l′$ after the transformation by matrix $M$ if the original equation of the line $l$ is $2x - 4y + 1 = 0$.

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