
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 \(a\), \(b\), \(c\) are positive integers such that \(a+b+c=39\) and \(\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=11\). Find the sum of all possible distinct values of \(a^2+b^2+c^2\).

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