
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
The total mass of  $100$  given weights with positive masses equals  $2S$ .  A natural number  $k$  is called *middle* if some  $k$  of the given weights have the total mass  $S$ .  Find the maximum possible number of middle numbers.

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