
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 coordinates \( (x ; y ; z) \) of point \( M \) are consecutive terms of a geometric progression, and the numbers \( x y, y z, x z \) in the given order are terms of an arithmetic progression, with \( z \geq 1 \) and \( x \neq y \neq z \). Find the smallest possible value of the square of the distance from point \( M \) to point \( N(1 ; 1 ; 1) \).

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