
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 \( x^3 + mx^2 + px + q = 0 \). Express the following determinant in terms of \( m, p, \) and \( q \):
\[
\begin{vmatrix}
a & 1 & 1 \\
1 & b & 1 \\
1 & 1 & c
\end{vmatrix}
\]

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