
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 $1 \frac{1}{2}$, $3 \frac{1}{4}$, $5 \frac{1}{8}$, $7 \frac{1}{16}$, ..., $(2n-1)+ \frac{1}{2^{n}}$, ... has a sum of the first $n$ terms $S_{n}$ equal to $n^{2}+1- \frac{1}{2^{n}}$.

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