
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 matrices $M = \left.\begin{matrix}1 & 0 \\ 0 & -1\end{matrix}\right.$ and $N = \left.\begin{matrix}1 & 2 \\ 3 & 4\end{matrix}\right.$.

(Ⅰ) Find the second-order matrix $X$ such that $MX = N$ holds;  
(Ⅱ) Find the eigenvalues of matrix $X$ and one eigenvector corresponding to each eigenvalue.

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