
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 $\left(a+kb,ka+b\right)$ (where $k$ is a constant and $k\neq 0$), then point $P'$ is called the "$k$-belonging derived point" of point $P$.

For example, the "$2$-belonging derived point" of $P(1,4)$ is $P'(1+2\times 4,2\times 1+4)$, which is $P'(9,6)$.

(1) The coordinates of the "$2$-belonging derived point" $P'$ of point $P(-1,6)$ are ________;

(2) If the coordinates of the "$3$-belonging derived point" $P'$ of point $P$ are $(6,2)$, then the coordinates of point $P$ are ________;

(3) If point $P$ is on the positive half-axis of the $x$-axis, and the point $P$'s "$k$-belonging derived point" is point $P'$, and the length of line segment $PP'$ is twice the length of line segment $OP$, find the value of $k$.

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