
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
Determine matrix $A$ and its inverse given that $A=\begin{pmatrix}3 & 3 \\ c & d\end{pmatrix}$, where $A$ has an eigenvector $\vec{\alpha}_1 = \begin{pmatrix}1 \\ 1\end{pmatrix}$ associated with eigenvalue 6, and an eigenvector $\vec{\alpha}_2 = \begin{pmatrix}3 \\ -2\end{pmatrix}$ associated with eigenvalue 1.

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