
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 \(x, y, z\), the matrix
\[
\begin{pmatrix} 
x & y & z \\ 
y & z & x \\ 
z & x & y 
\end{pmatrix}
\]
is not invertible. Find all possible values of
\[
\frac{x}{y + z} + \frac{y}{x + z} + \frac{z}{x + y}.
\]

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