
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
Determine the sum of all possible distinct values of \( p^2+q^2+r^2 \) given that \( p \), \( q \), and \( r \) are positive integers such that \( p+q+r=30 \) and \( \gcd(p,q)+\gcd(q,r)+\gcd(r,p)=12 \).

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