
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 $x$, $y$, $z$ are positive integers such that $x+y+z=30$ and $\gcd(x,y)+\gcd(y,z)+\gcd(z,x)=12$, calculate the sum of all possible distinct values of $x^2+y^2+z^2$.

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