
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 $A= \begin{pmatrix} 1 & 2 \\ -1 & 4 \end{pmatrix}$．  
(1) Find the inverse matrix $A^{-1}$；  
(2) Find the eigenvalues $\lambda_1$, $\lambda_2$ and a corresponding eigenvector $\overrightarrow {\alpha_{1}}$, $\overrightarrow {\alpha_{2}}$．

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