
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
Given a quadratic function $y=x^{2}-6ax+9(a$ is a constant).<br/>$(1)$ If the graph of this function passes through the point $\left(2,7\right)$, find the values of $a$ and the coordinates of the vertex of the graph;<br/>$(2)$ Under the conditions of $(1)$, when $-1\leqslant x \lt 3$, find the range of values of $y$;<br/>$(3)$ When $x\geqslant 3$, $y$ increases as $x$ increases. $P(x_{1}$,$y_{1})$ and $Q(x_{2}$,$y_{2})$ are two points on the graph of this function, and for any $3a-2\leqslant x_{1}\leqslant 5$, $3a-2\leqslant x_{2}\leqslant 5$, $y_{1}$ and $y_{2}$ always satisfy $y_{1}-y_{2}\leqslant 9a^{2}+20$. Find the range of values of $a$.

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