
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  $x$ ,  $y$ , and  $z$  are real numbers such that  $ x^2 + z^2 = 1 $  and  $ y^2 + 2y \left( x + z \right) = 6 $ , find the maximum value of  $ y \left( z - x \right) $ .

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