
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 a sequence of sums $S_n$, with $S_1 = 1 \cdot C_n^0 + 2 \cdot C_n^1 = 3 \cdot 2^1$, $S_2 = 1 \cdot C_n^0 + 2 \cdot C_n^1 + 3 \cdot C_n^2 = 4 \cdot 2^2$, $S_3 = 1 \cdot C_n^0 + 2 \cdot C_n^1 + 3 \cdot C_n^2 + 4 \cdot C_n^3 = 5 \cdot 2^3$, and so on, infer a general conclusion for $S_n = 1 \cdot C_n^0 + 2 \cdot C_n^1 + 3 \cdot C_n^2 + \dots + n \cdot C_n^n$.

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