
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 increasing arithmetic sequence, where $a_3=5$, and $a_1$, $a_2$, $a_5$ form a geometric sequence.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ Let $b_n= \frac {1}{(a_n+1)(a_{n+1}+1)}$, and denote the sum of the first $n$ terms of the sequence $\{b_n\}$ as $T_n$, find the smallest positive integer $m$ such that $T_n < \frac {m}{5}$ holds.

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