
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
Solve the following system of equations:

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

What is the condition for this system to have a unique solution? What cases arise if

$(1)$

$(2)$

$$
\begin{aligned}
& a+b=0 \\
& a-b=0 \\
& a=0
\end{aligned}
$$

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