
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 graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ passes through the points $A\left(-3,m\right)$ and $B\left(-1,n\right)$.<br/>$(1)$ When $m=n$, find the length of the line segment $AB$ and the value of $h$;<br/>$(2)$ If the point $C\left(1,0\right)$ also lies on the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$, and $m \lt 0 \lt n$,<br/>① find the abscissa of the other intersection point of the graph of the quadratic function $y=a\left(x-h\right)^{2}+k\left(a \lt 0\right)$ with the $x$-axis (expressed in terms of $h$) and the range of values for $h$;<br/>② if $a=-1$, find the area of $\triangle ABC$;<br/>③ a line passing through point $D(0$,$h^{2})$ perpendicular to the $y$-axis intersects the parabola at points $P(x_{1}$,$y_{1})$ and $(x_{2}$,$y_{2})$ (where $P$ and $Q$ are not coincident), and intersects the line $BC$ at point $(x_{3}$,$y_{3})$. Is there a value of $a$ such that $x_{1}+x_{2}-x_{3}$ is always a constant? If so, find the value of $a$; if not, explain why.

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