
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

Pedrinho chose 8 distinct numbers between 1 and 11 and wrote them in a certain order. Joãozinho, seeing the numbers that Pedrinho wrote, noticed the following curious fact: if we take the average of the first $n$ numbers written by Pedrinho, for $n=1, \ldots, 8$, the result is always an integer. In other words, if we take the average of the first two numbers, the first three numbers, the first four numbers, and so on, all these averages will be integers.

What are the possible sequences of numbers that Pedrinho wrote? (Hint: first find out what the possible sums for the 8 numbers are, and then try to discover the chosen numbers from back to front.)

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