
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 $xOy$, the point $\left(2,m\right)$ lies on the parabola $y=ax^{2}+bx+c\left(a \gt 0\right)$, where the axis of symmetry of the parabola is $x=t$.  
$(1)$ When $m=c$, find the value of $t$;  
$(2)$ Points $(-1$,$y_{1})$ and $(3$,$y_{2})$ lie on the parabola. If $c \lt m$, compare the values of $y_{1}$ and $y_{2}$, and explain the reason.

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