
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
In integer division, there are only two cases: divisible and non-divisible. When it is not divisible, a remainder will be produced. Now we use integer division to study a type of number called "difference number." Definition: For a natural number, if the number leaves a remainder of $6$ when divided by $7$ and a remainder of $4$ when divided by $5$, then it is called a "difference number." For example, $34\div 7=4\ldots 6$, $34\div 5=6\ldots 4$, so $34$ is a "difference number"; $27\div 7=3\ldots 6$, $27\div 5=5\ldots 2$, so $27$ is not a "difference number".<br/>$(1)$ Determine whether $69$ and $97$ are "difference numbers" and explain the reasons;<br/>$(2)$ Find all "difference numbers" greater than $500$ and less than $600$.

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