
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
Suppose $x$, $y$, and $z$ are positive integers such that \[x + y + z = 30\] and \[\gcd(x, y) + \gcd(y, z) + \gcd(z, x) = 12.\] What is the sum of all possible distinct values of $x^2 + y^2 + z^2$?
**A) 240** 
**B) 360** 
**C) 432** 
**D) 504** 
**E) 730**

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