
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
Given a fraction $\frac{a}{b}$, not necessarily in the simplest form, where $a$ and $b$ are positive integers whose sum is 18, find the number of distinct integers that can be written as the sum of two, not necessarily different, such special fractions.

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