
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 $n$ be a given positive integer, and let $M=\left\{\frac{1}{2^n}, \frac{1}{2^{n+1}}, \ldots, \frac{1}{2^{2n}}\right\}$. Denote all the subsets of $M$ as $M_1, M_2, \ldots, M_t$, and for $1 \leq i \leq t$, let $S(M_i)$ denote the sum of all elements in $M_i$, where we define $S(\varnothing)=0$. Then,  
①For $n=2$, the sum $S(M_1)+S(M_2)+\ldots+S(M_8)$ equals ____.  
②For $n \in \mathbb{N}^*$, the sum $S(M_1)+S(M_2)+\ldots+S(M_t)$ equals ____.

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