
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 an integer number of grams, with the total weight of all the weights being 500 grams. This set is called a correct set if any body, having a weight expressed as an integer number of grams from 1 to 500, can be balanced by some quantity of weights from the set, and only in a unique way. (This means the body is placed on one pan of the scale, and the weights on the other pan; 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 correct set in which not all weights are one gram each.

b) How many different correct sets exist?

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

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