
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
In the sequence $\left\{ a_n \right\}$, $a_1=2$, $a_{n+1}=a_n+cn$ ($c$ is a non-zero constant, $n=1,2,3,\cdots$), and $a_1$, $a_2$, $a_3$ form a geometric sequence.

$(1)$ Find the value of $c$;

$(2)$ Find the general formula for $\left\{ a_n \right\}$.

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