
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 function \( f: \mathbf{N} \rightarrow \mathbf{N} \) defined as follows:
$$
f(x)=\left\{\begin{array}{ll}
\frac{x}{2}, & x \text{ is even}; \\
\frac{x+7}{2}, & x \text{ is odd.}
\end{array}\right.
$$

The number of elements in the set \( A=\{x \in \mathbf{N} \mid f(f(f(x)))=x\} \) is .

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