Define the sequences $a_n$, $b_n$, and $c_n$ as follows:  
- $a_0 = k$, $b_0 = 4$, $c_0 = 1$.

For each $n$:
- If $a_n$ is even, then:
  - $a_{n+1} = \frac{a_n}{2}$,
  - $b_{n+1} = 2b_n$,
  - $c_{n+1} = c_n$.
- If $a_n$ is odd, then:
  - $a_{n+1} = a_n - \frac{b_n}{2} - c_n$,
  - $b_{n+1} = b_n$,
  - $c_{n+1} = b_n + c_n$.

Find the number of positive integers $k < 1995$ such that for some $n$, $a_n = 0$.

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