
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 situation where Harold, Tanya, and Ulysses paint a long fence. Harold starts with the first picket and paints every \( h \)th picket; Tanya starts with the second picket and paints every \( t \)th picket; Ulysses starts with the third picket and paints every \( u \)th picket. Define the integer \( 100h + 10t + u \) as paintable when the trio \( (h, t, u) \) results in every picket being painted exactly twice. Find the sum of all paintable integers.

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