
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
Given $n \in \mathbb{N}^*$, in the expansion of $(x+2)^n$, the coefficient of the second term is $\frac{1}{5}$ of the coefficient of the third term.  
(1) Find the value of $n$;  
(2) Find the term with the maximum binomial coefficient in the expansion;  
(3) If $(x+2)^n=a_0+a_1(x+1)+a_2(x+1)^2+\ldots+a_n(x+1)^n$, find the value of $a_0+a_1+\ldots+a_n$.

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