
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
A function \( f(n) \) for positive integer values is given by
\[
\begin{aligned}
f(1) & = 1, \quad f(3) = 3, \\
f(2 n) & = f(n), \\
f(4 n + 1) & = 2 f(2 n + 1) - f(n), \\
f(4 n + 3) & = 3 f(2 n + 1) - 2 f(n),
\end{aligned}
\]
for all positive integers \( n \). Determine with proof the number of positive integers less than or equal to 1988 for which \( f(n) = n \).

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