
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= \begin{bmatrix} 1 & a \\ 3 & b\\end{bmatrix}\\) has an eigenvalue \\(\lambda\_{1}=-1\\), and its corresponding eigenvector is \\(\overrightarrow{e}= \begin{bmatrix} \overset{1}{-1}\\end{bmatrix}\\), find the inverse of matrix \\(M\\).

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