
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 $a\_1$, $a\_3$, $a\_13$ form a geometric sequence.
(I) Find the general term formula of the sequence $\{a\_n\}$.
(II) Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. Does there exist a positive integer $n$ such that $S\_n > 40n+600$? If so, find the smallest value of $n$; if not, explain why.

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