
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 constant $k$, the matrix
\[
\begin{pmatrix} a & kb & kc \\ kb & c & ka \\ kc & ka & b \end{pmatrix}
\]
is not invertible. Determine all possible values of
\[
\frac{a}{kb + kc} + \frac{b}{ka + c} + \frac{c}{ka + b}.
\]

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