
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
A set consisting of at least two distinct positive integers is called *centenary* if its greatest element is  $100$ . We will consider the average of all numbers in a centenary set, which we will call the average of the set. For example, the average of the centenary set  $\{1, 2, 20, 100\}$  is  $\frac{123}{4}$  and the average of the centenary set  $\{74, 90, 100\}$  is  $88$ . Determine all integers that can occur as the average of a centenary set.

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