
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

Consider a set of weights, each weighing a whole number of grams, with the total weight of all the weights being 200 grams. Such a set is called proper if any object with a weight expressed in whole grams from 1 to 200 grams can be balanced with some number of weights from the set, in a unique way (the object is placed on one side of the scale, the weights on the other; two ways of balancing that differ only by replacing some weights with others of the same weight are considered identical).

a) Provide an example of a proper set where not all weights are one gram.

b) How many different proper sets exist?

(Two sets are different if some weight appears a different number of times in these sets.)

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