
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 the digits of a natural number are arranged from the highest place value to the lowest place value and the resulting number is the same as when the digits are arranged from the lowest place value to the highest place value, we call such a natural number a "harmonious number." For example, the natural number $12321$, when the digits are arranged from highest to lowest place value, is $1$, $2$, $3$, $2$, $1$, and when arranged from lowest to highest place value, it is still $1$, $2$, $3$, $2$, $1$, therefore $12321$ is a "harmonious number." Similarly, $22$, $545$, $3883$, $345543$, $\ldots$, are all "harmonious numbers."<br/>$(1)$ Please write down $3$ four-digit "harmonious numbers": ______;<br/>$(2)$ Let the digit in the units place of a four-digit "harmonious number" be $a$, and the digit in the tens place be $b$. Can you guess whether any four-digit "harmonious number" can be divisible by $11$? Please explain your reasoning.

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