
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 an arithmetic sequence $\left\{a_n\right\}$ satisfies: $a_1=2$, and $a_1$, $a_2$, $a_3$ form a geometric sequence.

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

(2) Let $S_n$ denote the sum of the first $n$ terms of the sequence $\left\{a_n\right\}$. Determine whether there exists a positive integer $n$ such that $S_n > 60n+800$. If it exists, find the minimum value of $n$; if not, explain why.

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