
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, the vertex of the parabola $y=ax\left(x-6\right)+1\left(a\neq 0\right)$ is $A$, and it intersects the $x$-axis at points $B$ and $C$ $(C$ is to the right of $B$).<br/>$(1)$ Determine whether the point $\left(0,1\right)$ lies on the parabola $y=ax\left(x-6\right)+1\left(a\neq 0\right)$, and explain the reason;<br/>$(2)$ If the distance from point $A$ to the $x$-axis is $5$, find the value of $a$;<br/>$(3)$ If the length of segment $BC$ is less than or equal to $4$, find the range of values for $a$.

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