
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 $\{a\_n\}$ is an increasing sequence, the sum of its first $n$ terms is $S\_n$, $a\_1 > 1$, and $10S\_n=(2a\_n+1)(a\_n+2)$, $n∈N^{+}$.
(I) Find the general term $a\_n$ of the sequence;
(II) Let $b\_n=a\_n- \frac {n-3}{2}$, if for any $n∈N^{+}$, the inequality $ \frac { \sqrt {5}m}{31}\leqslant (1+ \frac {1}{b\_1})(1+ \frac {1}{b\_2})…(1+ \frac {1}{b\_n})- \frac {1}{ \sqrt {2n+3}}$ always holds, find the maximum value of the positive integer $m$.

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