
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 scenario where Harold, Tanya, and Ulysses are tasked with painting a very long picket 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; and Ulysses starts with the fourth picket and paints every $u$th picket. Define a positive integer $100h+10t+u$ as paintable if the triple $(h,t,u)$ of positive integers results in every picket being painted exactly once. Find the sum of all paintable integers.

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