
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
Harold, Tanya, and Ulysses paint a long corridor with numbered tiles.
- Harold starts with the first tile and paints every \( h \)th tile.
- Tanya starts with the third tile and paints every \( t \)th tile.
- Ulysses starts with the fifth tile and paints every \( u \)th tile.
Define the positive integer \( 100h + 10t + u \) as paintable when the triple \( (h, t, u) \) of positive integers allows every tile to be painted exactly once by only one of them. Determine the sum of all paintable integers.

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