
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 $S_{n}$ is the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$, $a_{5}=2$, and $a_{n-1}+a_{n+1}=a_{5}a_{n}$ $(n\geqslant 2)$, and $a_{3}$ is the geometric mean between $a_{1}$ and $- \dfrac {8}{5}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If $a_{1}$ is an integer, and $b_{n}= \dfrac {n}{(2S_{n}+23n)(n+1)}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, $T_{n}$.

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