
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 parabola $y=ax^{2}+bx+c\left(a \gt 0\right)$ has the axis of symmetry $x=t$, and two different points $\left(3,m\right)$ and $\left(t+1,n\right)$ lie on the parabola.<br/>$(1)$ If $m=n$, find the value of $t$;<br/>$(2)$ If $n \lt m \lt c$, find the range of values for $t$.

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