
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
Suppose that $\{a\_n\}$ is an arithmetic sequence, and $S\_n$ is the sum of its first $n$ terms. Given that $S\_7=7$ and $S\_{15}=75$,

1. Find the general term formula of the sequence $\{a\_n\}$.
2. Let $b\_n=2a\_n+5$ and $T\_n= \frac {1}{b\_1b\_2}+ \frac {1}{b\_2b\_3}+…+ \frac {1}{b\_nb\_{n+1}}$. Does there exist a smallest positive integer $m$ such that $T\_n < \frac {m}{4}$ always holds for all $n∈N^{*}$? If so, find the value of $m$; if not, explain the reason.

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