
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
If  $x, y, z \in \mathbb{R}$  are solutions to the system of equations  $$ \begin{cases}
x - y + z - 1 = 0
xy + 2z^2 - 6z + 1 = 0
\end{cases} $$  what is the greatest value of  $(x - 1)^2 + (y + 1)^2$ ?

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