
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
While waiting for customers, a watermelon seller weighed 20 watermelons (each weighing 1 kg, 2 kg, 3 kg, ..., 20 kg) by balancing a watermelon on one side of the scale with one or two weights (possibly identical) on the other side of the scale. The seller recorded the weights he used on a piece of paper. What is the minimum number of different numbers that could appear in his notes 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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