
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
Count the total number of distinct $p$-arithmetic fractional-linear functions.

Denote $f(f(x))$ as $f_{2}(x)$. Similarly, define fractional-linear functions $f_{\mathrm{B}}(x), f_{1}(x), \ldots, f_{2}(x)$, using:

$$
\begin{aligned}
f_{3}(x) & = f(f_{2}(x)), \\
f_{4}(x) & = f(f_{\mathrm{B}}(x)), \\
\cdots & \cdot \\
f_{n}(x) & = f(f_{n-1}(x)).
\end{aligned}
$$

For example,

$$
\begin{aligned}
& f(x) = \frac{x+1}{x}, \\
& f_{2}(x) = \frac{\frac{x+1}{x}+1}{\frac{x+1}{x}} = \frac{2x+1}{x+1}, \\
& f_{3}(x) = \frac{\frac{2x+1}{x+1}+1}{\frac{2x+1}{x+1}} = \frac{3x+1}{2x+1}.
\end{aligned}
$$

The actions of all functions $f_{n}(x)$ can be traced on a scheme constructed for the function $f(x)$. The action of the function $f(x)$ is that each point of its scheme transitions to the next one, taking one step in the direction of the arrows. When applying the function $f_{2}(x)$, each point on the scheme of $f(x)$ takes two steps, ending up at the second point. Generally, when applying $f_{n}(x)$, each point on the scheme of $f(x)$ takes $n$ steps in the direction of the arrows, jumping over $n-1$ points to reach the $n$-th point. This allows us to build the schemes for the functions $f_{2}(x), f_{3}(x), \ldots$, based on the scheme for $f(x)$ without calculations.

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