IMO 1988 Problem 3

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

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

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