
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 symmetric matrix
\[\begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\]
Determine all possible values of
\[\frac{x}{y + z} + \frac{y}{z + x} + \frac{z}{x + y}\]
given that the matrix is not invertible.

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