
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 $xOy$, points $M(2,m)$ and $N(4,n)$ lie on the parabola $y=ax^{2}+bx$ where $a \gt 0$. <br/>$(1)$ If $m=n$, find the axis of symmetry of the parabola; <br/>$(2)$ Given that point $P(-1,p)$ lies on the parabola, and the axis of symmetry of the parabola is $x=t$. If $mn \lt 0$, and $m \lt p \lt n$, find the range of values for $t$.

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