
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
Suppose  $\{a_1,a_2,\ldots,a_7\}$  is a set of pair-wisely different positive integers. If  $a_1,2a_2,\ldots,7a_7$  can form an arithmetic series (in this order), find the smallest positive value of  $|a_7-a_1|$ .

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