
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
For real numbers $a,$ $b,$ and $c,$ the matrix
\[\begin{pmatrix} a^2 & b^2 & c^2 \\ b^2 & c^2 & a^2 \\ c^2 & a^2 & b^2 \end{pmatrix}\] is not invertible. List all possible values of
\[\frac{a^2}{b^2 + c^2} + \frac{b^2}{a^2 + c^2} + \frac{c^2}{a^2 + b^2}.\]

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