
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 Cartesian coordinate system, the expression of the graph of a quadratic function is $y=ax^{2}+\left(a+1\right)x+b$, where $a-b=4$.  
$(1)$ If the graph of this function passes through the point $\left(1,3\right)$, find the expression of this quadratic function.  
$(2)$ If $(x_{1},y_{1})(x_{2},y_{2})$ are two different points on the graph of this quadratic function, and when $x_{1}+x_{2}=2$, $y_{1}=y_{2}$, find the value of $a$.  
$(3)$ If the point $\left(-1,t\right)$ is on the graph of this quadratic function, and when $x\geqslant -1$, $y$ increases as $x$ increases, find the range of $t$.

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