
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 the set $A=\{x|1 < \log_{2}x < 3, x\in\mathbb{N}^{*}\}$, $B=\{4,5,6,7,8\}$.  
$(1)$ How many different three-digit numbers can be formed by taking out $3$ different elements from $A\cup B$?  
$(2)$ How many distinct natural numbers greater than $4000$, which do not have repeating digits, can be formed by taking out $1$ element from set $A$ and $3$ elements from set $B$?

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