
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
When $x \in \mathbb{R}$, and $|x| < 1$, the following expression holds: $1+x+x^{2}+…+x^{n}+…= \dfrac{1}{1-x}$.  
Integrating both sides, we get: $\int_{0}^{\frac{1}{2}}1dx+ \int_{0}^{\frac{1}{2}}xdx+ \int_{0}^{\frac{1}{2}}x^{2}dx+…+ \int_{0}^{\frac{1}{2}}x^{n}dx+…= \int_{0}^{\frac{1}{2}} \dfrac{1}{1-x}dx$.  
Thus, we obtain the following equation: $1\times \dfrac{1}{2}+ \dfrac{1}{2}\times\left( \dfrac{1}{2}\right)^{2}+ \dfrac{1}{3}\times\left( \dfrac{1}{2}\right)^{3}+…+ \dfrac{1}{n+1}\times\left( \dfrac{1}{2}\right)^{n+1}+…=\ln 2$.  
Based on the mathematical ideas and methods contained in the above materials, calculate:  
$C_{n}^{0}\times \dfrac{1}{2}+ \dfrac{1}{2} C_{n}^{1}\times\left( \dfrac{1}{2}\right)^{2}+ \dfrac{1}{3} C_{n}^{2}\times\left( \dfrac{1}{2}\right)^{3}+…+ \dfrac{1}{n+1} C_{n}^{n}\times\left( \dfrac{1}{2}\right)^{n+1}=$\_\_\_\_\_\_.

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