
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 $\{a\_n\}$ that satisfies: $a\_1=2$, and $a\_1$, $a\_2$, $a\_5$ form a geometric sequence.

1. Find the general term formula of the sequence $\{a\_n\}$.
2. Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. Determine whether there exists a positive integer $n$ such that $S\_n > 60n+800$. If it exists, find the minimum value of $n$; otherwise, explain the reason.

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