
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
\[
\mathbf{B} = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 2 \\ 3 & 2 & 1 \end{pmatrix}.
\]
Determine constants $a$, $b$, and $c$ such that
\[
\mathbf{B}^3 + a \mathbf{B}^2 + b \mathbf{B} + c \mathbf{I} = \mathbf{0},
\]
where $\mathbf{I}$ and $\mathbf{0}$ are the $3 \times 3$ identity matrix and zero matrix, respectively. Enter the ordered triple $(a,b,c)$.

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