
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 $a$ and $b \in R$, the matrix $A=\begin{bmatrix} a & b \\ 1 & 4 \\ \end{bmatrix}$. If a matrix $A$ has an eigenvector $\alpha_{1}=\begin{bmatrix} 3 \\ -1 \\ \end{bmatrix}$ corresponding to the eigenvalue $1$, and an eigenvector $\alpha_{2}=\begin{bmatrix} 1 \\ 1 \\ \end{bmatrix}$ corresponding to the eigenvalue $5$, find the matrix $A$ and write its inverse matrix.

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