
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
Given the set \( M = \{ x \mid 1 \leqslant x \leqslant 10, x \in \mathbf{N} \} \), for its non-empty subset \( A \), multiply each element \( k \) in \( A \) by \((-1)^{k}\) and then sum them (e.g., if \( A = \{1, 3, 6\} \), the sum is \((-1) \cdot 1 + (-1)^3 \cdot 3 + (-1)^6 \cdot 6 = 2\)). Find the total sum of these sums for all non-empty subsets of \( M \).

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