
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 \( f \) be a function defined on the set of positive rational numbers such that \( f(a \cdot b) = f(a) + f(b) \) for all positive rational numbers \( a \) and \( b \). If \( f(p) = p \) for every prime number \( p \), determine for which of the following numbers \( x \) is \( f(x) < 0 \)?
1. \(\frac{19}{45}\)
2. \(\frac{23}{36}\)
3. \(\frac{13}{25}\)
4. \(\frac{7}{18}\)
5. \(\frac{31}{14}\)
A) \(\frac{19}{45}\)  
B) \(\frac{23}{36}\)  
C) \(\frac{13}{25}\)  
D) \(\frac{7}{18}\)  
E) \(\frac{31}{14}\)

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