
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
Given two points $A$ and $B$ on the number line corresponding to $-2$ and $4$ respectively, point $P$ is a moving point on the number line corresponding to $x$.  
$(1)$ If the distance from point $P$ to points $A$ and $B$ are equal, find the number corresponding to point $P$;  
$(2)$ Does there exist a point $P$ on the number line such that the sum of the distances from point $P$ to points $A$ and $B$ is $8$? If yes, find the value of $x$; if not, explain the reason;  
$(3)$ Given that point $M$ on the number line starts moving leftward from point $A$ at a speed of $1$ unit length per second, and at the same time point $N$ starts moving leftward from point $B$ at a speed of $3$ unit lengths per second. After $t$ seconds, one of the points $M$, $N$, or $O$ (where $O$ is the origin) is exactly equidistant from the other two points. Find the value of $t$.

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