
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 a quadratic function $y=ax^{2}+bx+t-1$, where $t \lt 0$.<br/>$(1)$ When $t=-2$,<br/>① If the graph of the quadratic function passes through the points $\left(1,-4\right)$ and $\left(-1,0\right)$, find the values of $a$ and $b$;<br/>② If $2a-b=1$, for any non-zero real number $a$, is there a line $y=kx+p\left(k\neq 0\right)$ that always intersects the graph of the quadratic function at two different points? If yes, find the expression of the line; if not, explain the reason;<br/>$(2)$ If point $A\left(-1,t\right)$ and $B\left(m,t-n\right)\left(m \gt 0,n \gt 0\right)$ are two points on the graph of the quadratic function, and $S_{\triangle AOB}=\frac{1}{2}n-2t$, when $-1\leqslant x\leqslant m$, point $A$ is the highest point on the graph of the function, find the range of values of $a$.

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