
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
\( S(n) \) represents the sum of the digits of the natural number \( n \), for example, \( S(123) = 1 + 2 + 3 = 6 \). If two different positive integers \( m \) and \( n \) satisfy
\[
\left\{\begin{array}{l} 
m < 100 \\
n < 100 \\
m > n \\
m + S(n) = n + 2S(m)
\end{array}, \text{ then we call } m \text{ and } n \text{ a pair } \langle m, n \rangle.
\right.
\]

How many such pairs \( \langle m, n \rangle \) exist?


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