
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
From the numbers \( 10 \cdot 120 \), extract the positive rational numbers with denominators less than 1000 that are less than 1. Reduce all these numbers to their simplest form. Then, add the numerator and denominator of each number. How many different sums can be obtained?

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