
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 constants $p$, $q$, and $r$, the matrix
\[\begin{pmatrix} p & 2q & r \\ q & r & p \\ r & p & q \end{pmatrix}\]
is not invertible. Determine all possible values of
\[\frac{p}{2q + r} + \frac{2q}{p + r} + \frac{r}{p + q}.\]

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