
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
Consider the cubic polynomial $p(x) = x^3 - (a^2 + b^2 + c^2)x^2 + (a^4 + b^4 + c^4)x - abc$ whose roots are real numbers, and the matrix
\[ \begin{pmatrix} a^2 & b & c \\ b^2 & c & a \\ c^2 & a & b \end{pmatrix} \]
is not invertible. List all possible values of 
\[ \frac{a^2}{b^2 + c} + \frac{b^2}{a^2 + c} + \frac{c^2}{a^2 + b^2} \]

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