
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 the vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(\frac{1}{n^2+n},a\_n)(n\in\mathbb{N}^*)$, if $\overrightarrow{a}\parallel\overrightarrow{b}$, let the sum of the first $n$ terms of the sequence $\{a\_n\}$ be $S\_n$. Determine the minimum value of $S\_n$.

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