
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 in terms of $x$: $y=(m-2)x^{2}-x-m^{2}+6m-7$ ($m$ is a constant).<br/>$(1)$ If the graph of this quadratic function passes through the point $A(-1,2)$,<br/>① find the value of $m$;<br/>② if the graph of this quadratic function intersects the $x$-axis at points $B$ and $C$ (point $C$ is on the left side), find the area of $\triangle ABC$.<br/>$(2)$ If the graph of this quadratic function intersects the $y$-axis at point $P$, find the maximum value of the ordinate of point $P$.

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