
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 a geometric sequence $\{a\_n\}$ with a common ratio not equal to $1$, let $S\_n$ denote the sum of its first $n$ terms. It is known that $a\_3$ is the arithmetic mean of $a\_1$ and $a\_2$, and $S\_4+a\_2= \frac {1}{2}$.

1. Find $a\_n$;
2. Given an arithmetic sequence $\{b\_n\}$ with the sum of its first $n$ terms denoted as $T\_n$, where $b\_1=a\_3$, and $T\_7=49$, find $\frac {1}{b\_1b\_2}+ \frac {1}{b\_2b\_3}+...+ \frac {1}{b\_nb\_{n+1}}$.

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