
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 a set $A_n = \{1, 3, 7, \ldots, 2^n - 1\}$ for the first $n$ terms ($n \in \mathbb{N}^*$). From the set $A_n$, any $k$ ($k=1, 2, 3, \ldots, n$) numbers are randomly selected, and the sum of all possible products of these $k$ numbers is denoted as $T_k$ (if only one number is selected, the product is defined as the number itself). Let $S_n = T_1 + T_2 + \ldots + T_n$. For example, when $n=1$, $A_1 = \{1\}$, $T_1 = 1$, $S_1 = 1$; when $n=2$, $A_2 = \{1, 3\}$, $T_1 = 1 + 3$, $T_2 = 1 \times 3$, $S_2 = 1 + 3 + 1 \times 3 = 7$. Find the expression for $S_n = \_\_\_\_\_\_$.

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