
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 how many integers  $k$  does the following system of equations has a solution other than  $a=b=c=0$  in the set of real numbers? \begin{align*} \begin{cases} a^2+b^2=kc(a+b), b^2+c^2 = ka(b+c), c^2+a^2=kb(c+a).\end{cases}\end{align*}

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