
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
For a point $P(a,b)$ in the plane rectangular coordinate system $xOy$, if the coordinates of point $P$ are $(a+kb, ka+b)$ (where $k$ is a constant and $k\neq 0$), then point $P'$ is called the "$k$-derived point" of point $P$. For example, the "$4$-derived point" of $P(1,2)$ is $P'(1+4\times 2, 4\times 1+2)$, which is $P'(9,6)$.

$(1)$ The coordinates of the "$2$-derived point" $P'$ of point $P(-2,3)$ are ______;
$(2)$ If the coordinates of the "$3$-derived point" $P'$ of point $P$ are $(9,11)$, find the coordinates of point $P$;
$(3)$ If point $P$ lies on the positive $y$-axis, and the "$k$-derived point" $P'$ of point $P$ satisfies that the distance from point $P'$ to the $y$-axis is not less than $5$ times the length of segment $OP$, then the range of values for $k$ is ______.

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