
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 \(\mathbf{N}^{*}\) be the set of positive integers. Define a function \(f\) on \(\mathbf{N}^{*}\) as follows:
1. \(f(1)=1, f(3)=3\);
2. For \(n \in \mathbf{N}^{*}\), the following holds:
\[
\begin{array}{l}
f(2 n) = f(n), \\
f(4 n + 1) = 2 f(2 n + 1) - f(n), \\
f(4 n + 3) = 3 f(2 n + 1) - 2 f(n).
\end{array}
\]

Question: How many \(n \in \mathbf{N}^{*}\) are there such that \(n \leq 1988\) and \(f(n) = n\)?

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