
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
\[\bold{B} = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 2 \\ 3 & 2 & 1 \end{pmatrix}.\]
There exist constants $p'$, $q'$, and $r'$ such that
\[\bold{B}^3 + p' \bold{B}^2 + q' \bold{B} + r' \bold{I} = \bold{0},\]
where $\bold{I}$ and $\bold{0}$ are the $3 \times 3$ identity matrix and zero matrix, respectively. Find the ordered triple $(p',q',r').$

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