
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
For how many pairs of sequences of nonnegative integers  $(b_1,b_2,\ldots, b_{2018})$  and  $(c_1,c_2,\ldots, c_{2018})$  does there exist a sequence of nonnegative integers  $(a_0,\ldots, a_{2018})$  with the following properties:



- For  $0\leq i\leq 2018,$   $a_i<2^{2018}.$
- For  $1\leq i \leq 2018, b_i=a_{i-1}+a_i$  and  $c_i=a_{i-1}|a_i$ ;

where  $|$  denotes the bitwise or operation?


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