
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 plane rectangular coordinate system $xOy$, the parabola $G: y=ax^{2}+ax+c(a$ and $c$ are constants and $a \lt c)$ passes through the point $A\left(1,0\right)$ and has the vertex at $B$.  
$(1)$ Express $c$ in terms of $a$.  
$(2)$ Determine the quadrant in which point $B$ lies and provide a justification.  
$(3)$ If the line $l: y=2x-b$ passes through point $A$ and intersects the parabola $G$ at another point $C$, and the area of $\triangle ABC$ is $\frac{{27+18\sqrt{2}}}{4}$, find the range of values of $y=ax^{2}+ax+c$ for $-1 \lt x \lt 1$.

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