
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 $A$ with eigenvalue $\lambda_1=1$ and a corresponding eigenvector $e_{1}= \begin{bmatrix} 1 \\ 1 \end{bmatrix}$, and eigenvalue $\lambda_2=2$ with a corresponding eigenvector $e_{2}= \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, try to find the matrix $A$ and its inverse matrix $A^{-1}$.

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