
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 $M = \{x \mid 1 \leq x \leq 4, x \in \mathbb{N}\}$, for its non-empty subset $A$, one can multiply each element $k$ in $A$ by $(-1)^k$ and then sum them up (for example, if $A = \{1, 2, 4\}$, the sum would be $(-1)^1 \cdot 1 + (-1)^2 \cdot 2 + (-1)^4 \cdot 4 = 5$). Calculate the total sum of these sums over all non-empty subsets of $M$.

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