
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
The positive integers  $a, b, c$  are such that  $$ gcd \,\,\, (a, b, c) = 1, $$    $$ gcd \,\,\,(a, b + c) > 1, $$    $$ gcd \,\,\,(b, c + a) > 1, $$    $$ gcd \,\,\,(c, a + b) > 1. $$  Determine the smallest possible value of  $a + b + c$ .

Clarification: gcd stands for greatest common divisor.

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