
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
Let \( 2S \) be the total weight of a certain set of weights. We will call a natural number \( k \) "average" if it is possible to select \( k \) weights from the set such that their total weight is equal to \( S \). What is the maximum number of average numbers that a set of 100 weights can have?

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