A function $f$ defined on the positive integers (and taking positive integer values) is given by:

$$\begin{align*}
f(1) &= 1, \\
f(3) &= 3, \\
f(2 \cdot n) &= f(n), \\
f(4 \cdot n + 1) &= 2 \cdot f(2 \cdot n + 1) - f(n), \\
f(4 \cdot n + 3) &= 3 \cdot f(2 \cdot n + 1) - 2 \cdot f(n),
\end{align*}$$

for all positive integers $n$. Determine with proof the number of positive integers $\leq 1988$ for which $f(n) = n$.

Let's think step by step and output the final answer within \boxed{}.
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