
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 $xOy$, for points $P\left(x,y\right)$ and $Q\left(x,{y'}\right)$, the following definition is given: if ${y'}=\left\{\begin{array}{l}{y(x≥0)}\\{-y(x＜0)}\end{array}\right.$, then point $Q$ is called the "connecting point" of point $P$. For example, the "connecting point" of point $\left(2,3\right)$ is $\left(2,3\right)$, and the "connecting point" of point $\left(-2,-3\right)$ is $\left(-2,3\right)$.<br/>$(1)$ If point ${A'}\left(-2,-6\right)$ is the "connecting point" of point $A$ on the line $y=kx\left(k\neq 0\right)$, find the value of $k$;<br/>$(2)$ If point ${B'}\left(m,2\right)$ is the "connecting point" of point $B$ on the line $y=x+1$, find the coordinates of point $B$;<br/>$(3)$ Given that the line $y=-2x+2$ intersects the $x$-axis at point $P$, and the "connecting point" of a point $C$ on this line is point ${C'}$. If the area of $\triangle PCC'$ is $18$, find the length of segment $CP$.

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