
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
Define the function \( f \) on positive integers such that 

\[ f(1) = 1, \quad f(3) = 3 \]

and 

\[ 
\begin{aligned}
f(2n) &= f(n), \\
f(4n+1) &= 2f(2n+1) - f(n), \\
f(4n+3) &= 3f(2n+1) - 2f(n)
\end{aligned}
\]

for every positive integer \( n \).

Determine the number of integers \( n \) satisfying \( 1 \leq n \leq 1988 \) for which \( f(n) = n \).

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