
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
Find  $2^{2006}$  positive integers satisfying the following conditions.
(i) Each positive integer has  $2^{2005}$  digits.
(ii) Each positive integer only has 7 or 8 in its digits.
(iii) Among any two chosen integers, at most half of their corresponding digits are the same.

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