
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 \(x\), \(y\), and \(z\), consider the matrix
\[
\begin{pmatrix} x+y & x & y \\ x & y+z & y \\ y & x & x+z \end{pmatrix}
\]
Determine whether this matrix is invertible. If not, list 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