
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
Using the equation $kC_{n}^{k}=nC_{n-1}^{k-1}(1\leqslant k\leqslant n,k,n\in N^{\*})$, simplify the equation $1\cdot C_{n}^{1}+2\cdot C_{n}^{2}2^{1}+n\cdot C_{n}^{n}2^{n-1}=nC_{n-1}^{0}+n\cdot C_{n-1}^{1}2^{1}+n\cdot C_{n}^{2}2^{2}+…+n\cdot C_{n-1}^{n-1}2^{n-1}=n(1+2)^{n-1}=n\cdot 3^{n-1}.$ The equation $kC_{n}^{k}=nC_{n-1}^{k-1}$ has several variants, such as $\frac {1}{k} C_{ n-1 }^{ k-1 }= \frac {1}{n}C_{n}^{k}$ and $kC_{n+1}^{k}=(n+1)C_{n}^{k-1}$ (obtained by substituting $n+1$ for $n$), etc. Similarly, simplify the equation: $C_{n}^{0}\cdot \frac {1}{5}+ \frac {1}{2} C_{ n }^{ 1 }\cdot ( \frac {1}{5})^{2}+ \frac {1}{3} C_{ n }^{ 2 }\cdot ( \frac {1}{5})^{3}+…+ \frac {1}{n+1} C_{ n }^{ n }\cdot ( \frac {1}{5})^{n+1}=$ \_\_\_\_\_\_.

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