
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 $A= \begin{bmatrix} 1 & 2 \\ c & d \end{bmatrix}$ (where $c$ and $d$ are real numbers). If the matrix $A$ has eigenvalues 2 and 3, with corresponding eigenvectors $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ respectively, find the inverse matrix $A^{-1}$.

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