
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

There is a set of 20 weights that can be used to measure any integer weight from 1 to 1997 g (the weights are placed on one pan of the scales, the weight being measured is placed on the other). What is the minimum possible weight of the heaviest weight in such a set if:

a) the weights in the set are all integers

b) the weights are not necessarily integers

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