
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
For a four-digit positive integer $N$, where the digits in each place are all different and not equal to zero, if the sum of the digits in the thousands and tens places is equal to the sum of the digits in the hundreds and ones places, and both sums are 9, then $N$ is called a "Phoenix Number." In this case, it is defined that $K(N) = \frac{N}{99}$. For example, in the number $2475$, $2+7=4+5=9$, so $2475$ is a "Phoenix Number," and $K(2475) = \frac{2475}{99} = 25$. Another example is $2375$, where $3+5 \neq 9$, so $2375$ is not a "Phoenix Number."

$(1)$ Find $K(5841)$.

$(2)$ For a "Phoenix Number" $N$ that is even, if we swap the digits in the thousands and tens places, and also swap the digits in the hundreds and ones places, we get a new "Phoenix Number" $N'$. If $3K(N) + 2K(N')$ is a multiple of 9, and the thousands digit of $N$ is not less than the hundreds digit, then all the "Phoenix Numbers" $N$ that satisfy these conditions are ______.

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