
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 $x \in \mathbb{R}$, $|x| < 1$, the following expression holds: $1+x+x^2+…+x^n+…= \frac{1}{1-x^n}$,  
Integrating both sides, we get:  
$$\int_{0}^{\frac{1}{2}} 1dx + \int_{0}^{\frac{1}{2}} xdx + \int_{0}^{\frac{1}{2}} x^2dx + … + \int_{0}^{\frac{1}{2}} x^ndx + … = \int_{0}^{\frac{1}{2}} \frac{1}{1-x}dx$$  
Thus, we obtain the following equation: $1 \times \frac{1}{2} + \frac{1}{2} \times \left(\frac{1}{2}\right)^2 + \frac{1}{3} \times \left(\frac{1}{2}\right)^3 + … + \frac{1}{n+1} \times \left(\frac{1}{2}\right)^{n+1} + … = \ln3 - \ln2$.  
Based on the mathematical ideas and methods contained in the above materials, calculate:  
$C_n^0 \times \frac{1}{3} + \frac{1}{2}C_n^1 \times \left(\frac{1}{3}\right)^2 + \frac{1}{3}C_n^2 \times \left(\frac{1}{3}\right)^3 + … + \frac{1}{n+1}C_n^n \times \left(\frac{1}{3}\right)^{n+1} = \boxed{\text{\_\_\_\_\_\_}}$.

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