
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 that the matrix $M$ has an eigenvalue $\lambda_1 = 8$ and the corresponding eigenvector $a_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, and the transformation corresponding to matrix $M$ maps the point $(-1, 2)$ to $(-2, 4)$:

(Ⅰ) Find the matrix $M$;  
(Ⅱ) If a line $l$ is mapped to line $l'$: $x-2y=4$ under the linear transformation corresponding to matrix $M$, find the equation of line $l$.

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