
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 cubic equation $x^3 + px^2 + qx + r = 0$. Express the determinant:
\[
\begin{vmatrix}
1 + a^2 & 1 & 1 \\
1 & 1 + b^2 & 1 \\
1 & 1 & 1 + c^2 
\end{vmatrix}
\]
in terms of $p$, $q$, and $r$.

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