
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 Cartesian coordinate system, if the horizontal coordinate and the vertical coordinate of point $P$ are equal, then point $P$ is called a harmonic point. For example, points $(1,1)$, $(-\frac{1}{2},-\frac{1}{2})$, $(-\sqrt{3},-\sqrt{3})$, $\ldots$ are all harmonic points.  
$(1)$ Determine whether there exists a harmonic point on the graph of the function $y=-\frac{4}{x}$;  
$(2)$ If the graph of the quadratic function $y=ax^{2}+6x+c (a\neq 0)$ has exactly one harmonic point at $(\frac{5}{2},\frac{5}{2})$.  
① Find the values of $a$ and $c$;  
② If the function $y=ax^2+6x+c+\frac{1}{4} (a\neq 0)$ has a minimum value of $-1$ and a maximum value of $3$ for $1\leqslant x\leqslant m$, find the range of real numbers for $m$.

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