
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
Let \( X = \{1,2, \cdots, 100\} \). For any non-empty subset \( M \) of \( X \), define the characteristic of \( M \), denoted as \( m(M) \), as the sum of the maximum and minimum elements of \( M \). Find the average value of the characteristics of all non-empty subsets of \( X \).

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