
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
You are given  $n \ge 2$  distinct positive integers. Let's call a pair of these integers *elegant* if their sum is an integer power of  $2$ . For every  $n$  find the largest possible number of elegant pairs.

*Proposed by Oleksiy Masalitin*

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