
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
We know that on the number line, points $A$ and $B$ represent rational numbers $a$ and $b$ respectively, and the distance between the two points is $AB=|a-b|$. Therefore, the geometric meaning of $|x-3|$ is the distance between the point representing the rational number $3$ and the point representing the rational number $x$ on the number line. Based on the above information, please use the number line to answer the following questions:  
$(1)$ If $|x-3|=5$, then $x=$______;  
$(2)$ If $|x-3|=|x+1|$, then $x=$______.

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