
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, let the quadratic function be $y=(x+a)(x-a-1)$, where $a>0$. Find:<br/>
$(1)$ The axis of symmetry of the quadratic function;<br/>
$(2)$ If the maximum value of the function is $4$ when $-1\leqslant x\leqslant 3$, find the coordinates of the vertex of this quadratic function;<br/>
$(3)$ For two points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$ on the parabola, if $y_{1}\neq y_{2}$ for $t \lt x_{1} \lt t+1$ and $t+2 \lt x_{2} \lt t+3$, find the range of values for $t$.

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