
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 $a$ and $b$ are two distinct real numbers with $a \lt b$, we define that for a function, if its independent variable $x$ and function value $y$ satisfy: when $a\leqslant x\leqslant b$, $ta\leqslant y\leqslant tb(t$ is a positive number), we call this function a "$t$ times function" on $a\leqslant x\leqslant b$. For example, for the proportional function $y=2x$, when $1\leqslant x\leqslant 3$, $2\leqslant y\leqslant 6$, then $y=2x$ is a "$2$ times function" on $1\leqslant x\leqslant 3$.  
$(1)$ Given that the inverse proportional function $y=\frac{{2023}}{x}$ is a "$t$ times function" on $1\leqslant x\leqslant 2023$, find $t$;  
$(2)$ When $k \gt 0$, does there exist a linear function $y=kx+2$ that is a "$2k$ times function" on $a\leqslant x\leqslant b$? If it exists, find the value of $k$; if not, explain the reason;  
$(3)$ If the quadratic function $y=x^{2}-4x-7$ is a "$5$ times function" on $a\leqslant x\leqslant b$, find the values of $a$ and $b$.

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