
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 any non-empty subset  $X$  of  $M=\{1,2,3,...,2021\}$ , let  $a_X$  be the sum of the greatest and smallest elements of  $X$ . Determine the arithmetic mean of all the values of  $a_X$ , as  $X$  covers all the non-empty subsets of  $M$ .

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