
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
Eleanor, Finn, and George decide to paint a long fence. 
Eleanor starts with the first post and paints every $e$ th post; 
Finn starts with the third post and paints every $f$ th post; 
George starts with the second post and paints every $g$ th post.
Define a positive integer $100e + 10f+ g$ as a paintable integer when the triple $(e, f, g)$ of positive integers results in each post being painted exactly once by only one of them. Find the sum of all such paintable integers.

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