
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 sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?

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