
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 $a,$ $b,$ and $c,$ and a real scalar $\lambda,$ consider the matrix
\[\begin{pmatrix} a + \lambda & b & c \\ b & c + \lambda & a \\ c & a & b + \lambda \end{pmatrix}.\] Determine all possible values of \[\frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{a + b}\] assuming the matrix is not invertible.

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