
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
The sequence $\{2^{n}-1\}$ forms the set $A_{n}=\{1,3,7,2^{n}-1\}(n\in\mathbb{N}^{\*})$. Take any $k(k=1,2,3,…,n)$ numbers from the set $A_{n}$, and let $T_{k}$ be the sum of all possible products of these $k$ numbers (if only one number is taken, the product is defined as the number itself). Define $S_{n}=T_{1}+T_{2}+…+T_{n}$. For example, when $n=1$, $A_{1}=\{1\}$, $T_{1}=1$, $S_{1}=1$; when $n=2$, $A_{2}=\{1,3\}$, $T_{1}=1+3$, $T_{2}=1×3$, $S_{2}=1+3+1×3=7$. Write out $S_{n}=$ \_\_\_\_\_\_\_.

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