
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 $xOy$, given the parabola $y=x^{2}-2tx+1$.  
$(1)$ Find the axis of symmetry of the parabola (expressed in terms of $t$);  
$(2)$ If points $M(t-2,m)$ and $N(t+3,n)$ lie on the parabola $y=x^{2}-2tx+1$, compare the values of $m$ and $n$;  
$(3)$ Let $P(x_{1},y_{1})$ and $Q(x_{2},y_{2})$ be any two points on the parabola $y=x^{2}-2tx+1$. If for $-1\leqslant x_{1} \lt 3$ and $x_{2}=3$, it holds that $y_{1}\leqslant y_{2}$, find the range of $t$;  
$(4)$ Let $P(t+1,y_{1})$ and $Q(2t-4,y_{2})$ be two points on the parabola $y=x^{2}-2tx+1$, and both satisfy $y_{1}\geqslant y_{2}$. Find the maximum value of $t$.

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