Function?

Step 1 ：Apply the given property $f(3x) = 3f(x)$ repeatedly to find $f(2001)$.

Step 2 ：Apply the second part of the definition of $f$.

Step 3 ：Find the smallest $x$ for which $f(x) = 186.$

Step 4 ：Determine the range of $f(x)$ in the interval $x \in [1, 3]$ and other intervals.

Step 5 ：If $f(x) = 186,$ then $3^k \ge 186,$ so $k \ge 5.$

Step 6 ：Search the interval $x \in [3^5, 3^6] = [243, 729].$

Step 7 ：Let $y = \frac{x}{3^5},$ and find $f(y) = \frac{186}{3^5} = \frac{186}{243}.$

Step 8 ：Solve the equation $1 - |y-2| = \frac{186}{243}$ for $y.$

Step 9 ：The smaller of the two solutions is $y = 2 - \frac{57}{243} = \frac{429}{243}.$

Step 10 ：Find the final answer $x = 3^5y = \boxed{429}.$