
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 $\left(1,m\right)$ and $\left(4,n\right)$ lie on the parabola $y=ax^{2}+bx+c\left(a \gt 0\right)$. Let the axis of symmetry of the parabola be the line $x=t$.  
$(1)$ If $3a+b=0$, compare the relationship between $m$ and $n$, and explain the reason;  
$(2)$ The point $(x_{0},m)(x_{0}\neq 1)$ lies on the parabola, if $m \lt c \lt n$, find the range of values for $t$ and $x_{0}$.

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