
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
After learning about absolute values, we know that $|a|$ represents the distance of the number $a$ on the number line from the origin. For example, $|5|$ represents the distance of the point corresponding to $5$ on the number line from the origin. Similarly, $|5| = |5-0|$, which means $|5-0|$ represents the distance between the points corresponding to $5$ and $0$ on the number line. Similarly, we have $|5-3|$ represents the distance between the points corresponding to $5$ and $3$; $|5+3| = |5-(-3)|$, so $|5+3|$ represents the distance between the points corresponding to $5$ and $-3$. Generally, if points $A$ and $B$ on the number line represent the rational numbers $a$ and $b$ respectively, then the distance between $A$ and $B$ can be represented as $|a-b|$.

Please answer the following questions based on the meaning of absolute value and the number line:

$(1)$ The distance between the points representing $2$ and $5$ on the number line is \_\_\_\_; the distance between the points representing $-2$ and $-5$ on the number line is \_\_\_\_; the distance between the points representing $1$ and $-3$ on the number line is \_\_\_\_;

$(2)$ If $|a+1|=2$, then $a=$\_\_\_\_; if $|a+2|+|a-1|=5$, then $a=$\_\_\_\_\_\_\_;

$(3)$ Try to find the minimum value of $|a+5|+|a-1|+|a-3|$.

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