
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
Let $M_n$ be the set of $n$-digit decimal fractions of the form 0.$a_1a_2\ldots a_n$ where $a_i$ can only be 0 or 1 for $i=1,2,\ldots,n-1$, and $a_n=1$. Let $T_n$ be the number of elements in $M_n$, and $S_n$ be the sum of all elements in $M_n$. Find $S_n$.

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