
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
If $n$ is a three-digit positive integer, and the unit digit of $n$ is greater than the tens digit, and the tens digit is greater than the hundreds digit, then $n$ is called a "three-digit increasing number" (such as $146$, $369$, $567$, etc).  
$(1)$ From the numbers $1$, $2$, $3$, $4$, $5$, choose any three numbers to form a three-digit increasing number, find the probability that this number is divisible by $5$.  
$(2)$ In a math fun activity, each participant needs to randomly select one number from all "three-digit increasing numbers" and can only select once. The scoring rules are as follows: if the product of the three digits of the selected "three-digit increasing number" is not divisible by $3$ and not divisible by $5$, the participant gets $0$ points; if it is divisible by $3$ or $5, but not divisible by $15$, the participant gets $1$ point; if it is divisible by $15$, the participant gets $2$ points. It is known that participant A participates in this activity, find the distribution table and mathematical expectation of A's score $X$.

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