
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 $a\_1=3$, and $a\_n= \frac{1}{2}a_{n-1}+1 (n\geqslant 2, n\in N^*)$, find the general term formula for the sequence $\{a\_n\}$.

A) $\frac{2^{n}+1}{2^{n-1}}$
B) $\frac{2^{n}-1}{2^{n-1}}$
C) $\frac{2^{n}+1}{2^{n+1}}$
D) $\frac{2^{n}-1}{2^{n+1}}$

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