
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
Find a set \( S \) of positive integers such that no two distinct subsets of \( S \) have the same sum. Your score will be \(\left\lfloor 20\left(\frac{2^n}{r} - 2\right)\right\rfloor\), where \( n \) is the number of elements in the set \( S \), and \( r \) is the largest element of \( S \) (assuming, of course, that this number is nonnegative).

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