
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, Ulysses, and Vanessa paint 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; 
Ulysses starts with the third picket and paints every $u$ th picket; 
Vanessa starts with the fourth picket and paints every $v$ th picket.
Call the positive integer $1000h+100t+10u+v$ paintable when the quadruple $(h,t,u,v)$ 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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