
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
Let \( m \) and \( n \) be two coprime natural numbers with \( n < m \). Compare the following two sums:
$$
\sum_{k=1}^{n}\left\lfloor k \cdot \frac{m}{n}\right\rfloor \text{ and } \sum_{k=1}^{m}\left\lfloor k \cdot \frac{n}{m}\right\rfloor.
$$

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