
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 function $f(x)=\frac{1}{2}+\frac{1}{x}$, the positive sequence $\{a\_n\}$ satisfies $a\_1=1$, $a\_n=f(\frac{1}{a_{n-1}})$ for all $n\in\mathbb{N}^*$ and $n\geq 2$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. For all $n\in\mathbb{N}^*$, find the sum $S\_n=\frac{1}{a\_1a\_2}+\frac{1}{a\_2a\_3}+\frac{1}{a\_3a\_4}+\dots+\frac{1}{a\_na_{n+1}}$.

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