
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
Consider a team of painters, Anna, Bob, and Charlie, working on a series of numbered posts. Anna begins painting at the first post and paints every $a$-th post, Bob begins at the second post and paints every $b$-th post, and Charlie starts at the fourth post and paints every $c$-th post. Define a number as "properly paintable" if it's of the form $100a + 10b + c$ and if the way $a$, $b$, and $c$ are set up leads to every post being painted exactly once. Find the sum of all the properly paintable numbers.

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