
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
Let 
\[ \mathbf{A} = \begin{pmatrix} 1 & 1 & 2 \\ 1 & 1 & 1 \\ 2 & 1 & 1 \end{pmatrix}. \]
Determine constants \(p\), \(q\), and \(r\) such that
\[ \mathbf{A}^3 + p \mathbf{A}^2 + q \mathbf{A} + r \mathbf{I} = 2\mathbf{I}, \]
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix and \(2\mathbf{I}\) is a matrix with all diagonal elements as 2.

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