
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
Let $a$, $b$, and $c$ be positive integers such that $a + b + c = 30$ and $\gcd(a,b) + \gcd(b,c) + \gcd(c,a) = 11$. Determine the sum of all possible distinct values of $a^2 + b^2 + c^2$.
A) 302
B) 318
C) 620
D) 391
E) 419

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