
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
The sequence $\{a\_n\}$ is defined by $a\_1=2$ and $a\_{n+1}=a\_n+cn$, where $c$ is a constant. Additionally, $a\_1,a\_2,a\_3$ form a geometric sequence with a common ratio not equal to $1$.

(I) Find the value of $c$.
(II) Find the general term formula for $\{a\_n\}$.

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