<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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 $M$. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
16
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 $M$.
test