
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
Julia, Kevin, and Lee are painting a long straight fence. 
Julia starts with the first picket and paints every $j$th picket; 
Kevin starts with the second picket and paints every $k$th picket; 
Lee starts with the third picket and paints every $l$th picket.
Define a positive integer $100j+10k+l$ as perfectly paintable when the triple $(j,k,l)$ of positive integers results in every picket being painted exactly once by only one of the painters. Find the sum of all perfectly paintable integers.

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