
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 the matrix $M=\begin{bmatrix}1 & a \\ 3 & b\end{bmatrix}$ with one eigenvalue $\lambda_{1}=-1$ and its corresponding eigenvector $e=\begin{bmatrix}1 \\ -1\end{bmatrix}$. Find the other eigenvalue and its corresponding eigenvector for matrix $M$.

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