
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 of which weighs an integer number of grams, and the total weight of all the weights is 200 grams. Such a set is called correct if any object, having a weight expressed as an integer number of grams from 1 to 200, can be balanced by some number of weights from the set, in a unique way (the object is placed on one side of the scales, the weights - on the other side; 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 correct set in which not all weights are one gram each.

b) How many different correct sets are there?

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

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