
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 $M={1,2,3,…,n}$ $(n∈N^{+})$, for any non-empty subset $A$ of $M$, define $f(A)$ as the maximum element in $A$. When $A$ takes all non-empty subsets of $M$, the corresponding sum of $f(A)$ is $S_{n}$. Find:
$①S_{3}=$ _______
$②S_{n}=$ _______ .

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