
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 are tasked with painting a long fence. 
Harold starts painting from the second picket and paints every $h$th picket.
Tanya starts from the first picket and paints every $t$th picket.
Ulysses starts from the fourth picket and paints every $u$th picket.
Define the integer $100h+10t+u$ as paint-worthy when the triple $(h,t,u)$ of positive integers ensures that each picket is painted exactly once. Find the sum of all paint-worthy integers.

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