
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 \( S(n) \) denote the sum of the digits of a natural number \( n \). For example, \( S(123) = 1 + 2 + 3 = 6 \). If two different positive integers \( m \) and \( n \) satisfy the following conditions:
\[ 
\begin{cases}
m < 100 \\
n < 100 \\
m > n \\
m + S(n) = n + 2S(m)
\end{cases}
\]
then \( m \) and \( n \) form a pair \( \langle m, n \rangle \). 

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

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