
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 that $n \in \mathbb{N}^*,$ the coefficient of the second term is $\frac{1}{5}$ of the coefficient of the third term in the expansion of $(x+2)^n$.
(1) Find the value of $n$;
(2) 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}
user