
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
The real numbers  $x$ ,  $y$ ,  $z$ , satisfy  $0\leq x \leq y \leq z \leq 4$ .  If their squares form an arithmetic progression with common difference  $2$ , determine the minimum possible value of  $|x-y|+|y-z|$ .

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