
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 sequences $\{a_n\}$ and $\{b_n\}$, it is known that $a_1=0$, $a_2=1$, $b_1=1$, $b_2=\frac{1}{2}$. The sum of the first $n$ terms of sequence $\{a_n\}$ is $S_n$, and the sum of the first $n$ terms of sequence $\{b_n\}$ is $T_n$, satisfying $S_n+S_{n+1}=n^2$ and $2T_{n+2}=3T_{n+1}-T_n$, where $n$ is a positive integer.

$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.

$(2)$ Is there any positive integer $m$, $n$ such that $\frac{T_{n+1}-m}{T_n-m} > 1+b_{m+2}$ holds? If yes, find all the ordered pairs of real numbers $(m,n)$ that satisfy the condition; if no, please explain why.

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