
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 sequences  $ a_n, b_n, c_n$  as follows.  $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$ .
 If  $ a_n$  is even then  $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$ ,  $ b_{n \plus{} 1} \equal{} 2b_n$ ,  $ c_{n \plus{} 1} \equal{} c_n$ . 
If  $ a_n$  is odd, then  $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$ ,  $ b_{n \plus{} 1} \equal{} b_n$ ,  $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$ . 
Find the number of positive integers  $ k < 1995$  such that some  $ a_n \equal{} 0$ .

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