
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
The sequence $\{a_n\}$ satisfies: $a_1=1$, $a_2=2$, $a_{n+2}= \dfrac {a_{n}+a_{n+1}}{2} (n\in\mathbb{N}^*)$. Let $b_n=a_{n+1}-a_{n}$,

(1) Find the general formula for the sequence $\{b_n\}$;

(2) Find the smallest positive integer $N$ such that for $n > N$, $|a_{n}- \dfrac {5}{3}| < \dfrac {2}{9n}$ always holds;

(3) The sequence $\{c_n\}$ satisfies $c_n= \dfrac {3}{2}|a_{n}- \dfrac {5}{3}|$, and the sum of the first $n$ terms of $\{c_n\}$ is $T_n$. Is there any positive integers $m$, $n$ such that $\dfrac {T_{n+1}-m}{T_{n}-m} > c_{m+2}$ holds? If it exists, find all the ordered real number pairs $(m,n)$ that satisfy the condition; if not, explain why.

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