
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 $a, b, c$, the matrix
\[
\begin{pmatrix} 
a & b & c & 0 \\ 
b & c & 0 & a \\ 
c & 0 & a & b \\ 
0 & a & b & c 
\end{pmatrix}
\]
is not invertible. Determine 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}
user