
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, and the total weight of all the weights is 200 grams. This set is called "correct" if any object weighing an integer number of grams from 1 to 200 can be balanced by some number of weights from the set in one and only one way (the object is placed on one side of the scales, and the weights on the other; two ways of balancing are considered the same if they differ only by replacing some weights with others of the same weight).

a) Provide an example of a correct set in which not all the weights are 1 gram.

b) How many different correct sets are there?

(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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