
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
In the study of subset problems of finite sets, a student used the sum of all elements in a subset as the object of study. Specifically, the sum of the elements in an empty set is recorded as \(0.\) For example, the subsets of the set \(\\{1,2\\}\) are \(\varnothing\), \(\\{1\\}\), \(\\{2\\}\), \(\\{1,2\\}\), respectively. The sum of all elements in each subset is recorded as \(S\_1\), \(S\_2\), \(S\_3\), \(S\_4\), so \(S\_1=0\), \(S\_2=1\), \(S\_3=2\), \(S\_4=3\), then \(S=S\_1+S\_2+S\_3+S\_4=0+1+2+3=6\); The subsets of the set \(\\{0,1,2\\}\) are \(\varnothing\), \(\\{0\\}\), \(\\{1\\}\), \(\\{2\\}\), \(\\{0,1\\}\), \(\\{0,2\\}\), \(\\{1,2\\}\), \(\\{0,1,2\\}\), respectively. The sum of all elements in each subset is recorded as \(S\_1\), \(S\_2\), \(S\_3\), …, \(S\_8\), so \(S=S\_1+S\_2+S\_3+…+S\_8=12\); … Through further research, he discovered a correct conclusion: The subsets of the set \(\\{a\_1,a\_2,…,a\_n\\}\) are \(\varnothing\), \(\\{a\_1\\}\), \(\\{a\_2\\}\), …, \(\\{a\_1,a\_2,…,a\_n\\}\), respectively. The sum of all elements in each subset is recorded as \(S\_1\), \(S\_2\), \(S\_3\), …, \({{S}\_{{{2}^{n}}}}\), then the value of \(S=S\_1+S\_2+S\_3+…+{{S}\_{{{2}^{n}}}}\) is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ .

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