
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 points $\left(2a+1,m\right)$ and $\left(b,n\right)$ are points on the parabola $y=ax^{2}-2a^{2}x+c\left(a\neq 0,c \gt 0\right)$. <br/>$(1)$ When $a=1$, find the axis of symmetry of the parabola and directly write the relationship between $m$ and $c$; <br/>$(2)$ If for any $2\leqslant b\leqslant 4$, $m \gt c \gt n$, find the range of values for $a$.

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