
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 $xOy$, $M(x_{1}, y_{1})$, $N(x_{2}, y_{2})$ are any two points on the parabola $y=ax^{2}+bx+c\left(a \gt 0\right)$, where the axis of symmetry of the parabola is $x=t$.  
$(1)$ If $y_{1}=y_{2}$ for $x_{1}=1$ and $x_{2}=2$, find the value of $t$;  
$(2)$ If $y_{1} \lt y_{2}$ for $0 \lt x_{1} \lt 1$ and $1 \lt x_{2} \lt 2$, find the range of values for $t$.

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