
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 \( S \) be the set of all rational numbers in the interval \(\left(0, \frac{5}{8}\right)\). For a fraction \(\frac{q}{p} \in S\) where \((p, q) = 1\), define the function \( f\left(\frac{q}{p}\right) = \frac{q+1}{p} \). The number of roots of \( f(x) = \frac{2}{3} \) in the set \( S \) is \(\qquad\).

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