
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 $A= \begin{bmatrix} a & b \\ 2 & 1\end{bmatrix}$, if the matrix $A$ has an eigenvector $\alpha_1= \begin{bmatrix} 1 \\ -1\end{bmatrix}$ corresponding to the eigenvalue $-1$, and an eigenvector $\alpha_2= \begin{bmatrix} 3 \\ 2\end{bmatrix}$ corresponding to the eigenvalue $4$. Find the matrix $A$ and write down the inverse matrix $A^{-1}$.

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