
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 the set
$$
\begin{array}{l}
M=\{(m, n) \mid m \text{ and } n \text{ are coprime, } m+n>2020, \\
\quad 0<m<n \leqslant 2020\}.
\end{array}
$$
Find the sum \( S=\sum_{(m, n) \in M} \frac{1}{mn} \).

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