
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
Given real numbers $a, b,$ and $c$, such that $a^3 + b^3 + c^3 = 3abc$, consider the matrix
\[
\begin{pmatrix} a & b & c \\ c & a & b \\ b & c & a \end{pmatrix}
\]
is not invertible. Find all possible values of
\[
\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}.
\]

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