
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 that the sequence $\{a\_n\}$ is an arithmetic sequence with a common difference $d$ not equal to zero, $\{b\_n\}$ is a geometric sequence, and the function $f(x) = b\_1x^2 + b\_2x + b\_3$ has a y-intercept of $-4$ and a maximum value of $a\_6 - \frac{7}{2}$.

1. Find the value of $a\_6$.
2. If $f(a\_2+a\_8) = f(a\_3+a\_11)$, find the general term formula for the sequence $\{b\_n\}$.
3. If $a\_2 = -\frac{7}{2}$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{a\_na\_{n+1}}\}$. Find $T\_n$.

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