
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 a sequence $\{a\_n\}$ with the first term $1$, and $f(n)=a\_1C\_n^1+a\_2C\_n^2+⋯+a\_kC\_n^k+⋯+a\_nC\_n^n (n∈N^∗).$

(1) If $\{a\_n\}$ is a constant sequence, find the value of $f(5)$;

(2) If $\{a\_n\}$ is a geometric sequence with common ratio $3$, find the analytical expression of $f(n)$;

(3) Determine whether the sequence $\{a\_n\}$ can be an arithmetic sequence such that $f(n)-1=(n-1)2^n$ holds for all $n∈N^∗$. If possible, find the general formula for the sequence $\{a\_n\}$; if not, explain the reason.

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