
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
Read the following information:<br/>① Two points $M$ and $N$ on the number line represent numbers $x_{1}$ and $x_{2}$, respectively. The distance between point $M$ and point $N$ is $|MN|$ and $|MN|=|x_{1}-x_{2}|$.<br/>② When three points $A$, $B$, and $C$ on the number line satisfy $|CA|=k|CB|\left(k \gt 1\right)$, then point $C$ is called "$A$ is $k$ related to $B$". For example, when points $A$, $B$, and $C$ represent numbers $0$, $1$, $2$, $|CA|=2|CB|$, so $C$ is "$A$ is $2$ related to $B$".<br/>Based on the above information, answer the following questions:<br/>Given that points $A$ and $B$ represent numbers $-3$ and $6$ on the number line, and the moving point $P$ represents the number $x$ on the number line:<br/>$(1)$ If point $P$ is "$A$ is $2$ related to $B$", then $x=$______；<br/>$(2)$ If $x$ satisfies $|x+2|+|x-1|=3$, and point $P$ is "$A$ is $k$ related to $B$", then the range of $k$ is______；<br/>$(3)$ If the moving point $P$ starts from point $A$ and moves to the right at a speed of $1$ unit per second, and at the same time the moving point $Q$ starts from point $B$ and moves to the left at a speed of $2$ units per second, when moving for $t$ seconds, point $Q$ is exactly "$P$ is $3$ related to $A$", find the value of $t$.

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