
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 quadratic function $y=x^{2}-2tx+3\left(t \gt 0\right)$:<br/>$(1)$ If its graph passes through the point $\left(2,1\right)$, what is the value of $t$?<br/>$(2)$ When $0\leqslant x\leqslant 3$, the minimum value of $y$ is $-2$, find the value of $t$;<br/>$(3)$ If $A\left(m-2,a\right)$, $B\left(4,b\right)$, $C\left(m,a\right)$ are all on the graph of this quadratic function, and $a \lt b \lt 3$. Find the range of values for $m$.

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