
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
What is the necessary and sufficient condition for the following system of equations:

$$
\begin{aligned}
& x=b y+c z \\
& y=c z+a x \\
& z=a x+b y
\end{aligned}
$$

to have a non-zero solution?

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