
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 system of equations for $x$, $y$, and $z$ given by

$$
\begin{aligned}
a & =\frac{a x+c y}{b z+1}, \\
b & =\frac{b x+y}{b z+1}, \\
c & =\frac{a z+c}{b z+1},
\end{aligned}
$$

assuming $a \lessgtr bc$.

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