
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
It is given that \( a, b \), and \( c \) are three positive integers such that
\[ a^{2} + b^{2} + c^{2} = 2011. \]

Let the highest common factor (HCF) and the least common multiple (LCM) of the three numbers \( a, b, \) and \( c \) be denoted by \( x \) and \( y \) respectively. Suppose that \( x + y = 388 \). Find the value of \( a + b + c \).

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