
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
In the arithmetic sequence $\{a_n\}$, $a_1+a_2=7$, $a_3=8$. Let $b_n= \frac {1}{a_{n}a_{n+1}}$, the sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$.  
(I) Find the general formula for the sequence $\{a_n\}$.  
(II) Find the sum of the first $n$ terms of the sequence $\{b_n\}$, $T_n$.  
(III) Does there exist positive integers $m$, $n$ ($1<m<n$) such that $T_1$, $T_m$, $T_n$ form a geometric sequence? If so, find all possible values of $m$ and $n$; if not, explain why.

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