
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
Imagine that Lucia, Miguel, and Nadia are painting a fence. Lucia starts with the first picket and paints every \( h \} th picket; Miguel starts with the third picket and paints every \( t \) th picket; and Nadia starts with the second picket and paints every \( u \) th picket. Define a number \( 100h+10t+u \) as fully paintable when the triple \( (h,t,u) \) of positive integers ensures that every picket is painted exactly once. Find the sum of all fully paintable numbers.

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