
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
The watermelon seller weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) one by one while waiting for customers, balancing a watermelon on one pan of the scale with one or two weights on the other pan (possibly identical). At the same time, the seller noted down which weights he used. What is the minimum number of different numbers that could have appeared in his records if the weight of each weight is an integer number of kilograms?

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