
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 the graph of the quadratic function $y=-x^{2}+2ax+2a$ passes through the point $A(-1,m)$.  
$(1)$ Fill in the blank: $m=$______;  
$(2)$ If the graph of the quadratic function passes through $B(3,-1)$, when $-2\leqslant x\leqslant 3$, find the maximum and minimum values of the quadratic function $y=-x^{2}+2ax+2a$;  
$(3)$ Draw a line $AC$ parallel to the x-axis through point $A$, intersecting the graph of the quadratic function at point $C$, where points $A$ and $C$ are not coincident. Express the length of segment $AC$ in terms of $a$;  
$(4)$ Given point $E(a-1,-1)$, point $F(2a+3,-1)$, when the graph of the quadratic function has exactly one common point with line segment $EF$, directly write down the range of values for $a$.

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