
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 of weights is considered, each of which weighs an integer number of grams, and the total weight of all the weights is 500 grams. Such a set is called proper if any object with a weight expressed in an integer number of grams from 1 to 500 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 the same).

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

b) How many different proper sets exist?

(Two sets are different if there is a weight that participates in these sets a different number of times.)

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