
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 \(a\), \(b\), \(c\) be the roots of the polynomial \(x^3 - ax^2 + px + q = 0\). Express
\[
\begin{vmatrix} a + 1 & 2 & 2 \\ 2 & b + 1 & 2 \\ 2 & 2 & c + 1 \end{vmatrix}
\]
in terms of \(a\), \(p\), and \(q\).

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