
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 combining numbers and shapes is an important method for solving mathematical problems. For example, $|3-1|$ can be represented as the distance between the points $3$ and $1$ on the number line, and $|3+1|$ or $|3-(-1)|$ represents the distance between the points $3$ and $-1$. The geometric meaning of the expression $|x-1|$ is the distance between the point corresponding to $x$ on the number line and the point corresponding to $1$. Similarly, $|x+2|=|x-(-2)|$, so the geometric meaning of $|x+2|$ is the distance between the point corresponding to $x$ on the number line and the point corresponding to $-2$. Based on the above discoveries, explore the following:<br/>$(1)$ Directly calculate $|8-(-2)|=$______；<br/>$(2)$ Using the number line, find all integers $x$ that satisfy the condition $|x-2|+|x+3|=5$ and calculate the sum of all these integers.<br/>$(3)$ Based on the exploration above, for any rational number $x$, does $|x+4|+|x-6|$ have a minimum value? If yes, write down the minimum value and explain the reason; if not, explain the reason.

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