
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 the function \( f:(0,1) \rightarrow \mathbf{R} \) be defined as
$$
f(x)=\left\{\begin{array}{l}
x, \text{ if } x \text{ is irrational; } \\
\frac{p+1}{q}, \text{ if } x=\frac{p}{q}, \text{ where } (p, q)=1 \text{ and } 0<p<q.
\end{array}\right.
$$

Find the maximum value of \( f(x) \) on the interval \( \left(\frac{7}{8}, \frac{8}{9}\right) \).

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