
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 an eigenvalue $\lambda=8$ and a corresponding eigenvector $\overrightarrow{e_{1}} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, and the transformation corresponding to matrix $M$ maps point $A(-1,2)$ to $A'(-2,4)$.
(1) Find the matrix $M$;
(2) Let line $l$ be transformed to line $m: x - y = 6$ under the transformation corresponding to $M^{-1}$. Find the equation of line $l$.

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