
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 $M= \begin{bmatrix} a & 1 \\ 3 & b \end{bmatrix}$, one of the eigenvectors corresponding to its eigenvalue $\lambda = -1$ is $\overrightarrow{e_{1}}= \begin{bmatrix} 1 \\ -3 \end{bmatrix}$.  
(1) Find the matrix $M$;  
(2) Suppose the equation of curve $C'$ obtained from the transformation of curve $C$ by matrix $M$ is $xy=1$, find the equation of curve $C$.

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