
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 plane rectangular coordinate system $xOy$, for point $P(x,y)$, if the coordinates of point $Q$ are $(x+ay, ax+y)$, then point $Q$ is called the "$a$-level follower point" of point $P$ (where $a$ is a constant and $a\neq 0)$. For example, the "$2$-level follower point" of point $P(1,4)$ is point $Q(1+2\times 4, 2\times 1+4)$, which means the coordinates of point $Q$ are $(9,6)$.
$(1)$ If the coordinates of point $P$ are $(-3,5)$, find the coordinates of its "$3$-level follower point";
$(2)$ If point $P(c+2,2c-1)$ is first moved $2$ units to the left, and then moved $3$ units up to get point $P_{1}$, and the "$-3$-level follower point" $P_{2}$ of point $P_{1}$ is on the coordinate axis, find the coordinates of point $P_{2}$.
$(3)$ If point $P$ is on the positive $x$-axis, the "$a$-level follower point" of point $P$ is point $P_{3}$, and the length of line segment $PP_{3}$ is twice the length of line segment $OP$, find the value of $a$.

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