
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 any four-digit number $m$, if the digits of $m$ are all non-zero and distinct, and the sum of the units digit and the thousands digit is equal to the sum of the tens digit and the hundreds digit, then this number is called a "mirror number". If we swap the units digit and the thousands digit of a "mirror number" to get a new four-digit number $m_{1}$, and swap the tens digit and the hundreds digit to get another new four-digit number $m_{2}$, let $F_{(m)}=\frac{{m_{1}+m_{2}}}{{1111}}$. For example, if $m=1234$, swapping the units digit and the thousands digit gives $m_{1}=4231$, and swapping the tens digit and the hundreds digit gives $m_{2}=1324$, the sum of these two four-digit numbers is $m_{1}+m_{2}=4231+1324=5555$, so $F_{(1234)}=\frac{{m_{1}+m_{2}}}{{1111}}=\frac{{5555}}{{1111}}=5$. If $s$ and $t$ are both "mirror numbers", where $s=1000x+100y+32$ and $t=1500+10e+f$ ($1\leqslant x\leqslant 9$, $1\leqslant y\leqslant 9$, $1\leqslant e\leqslant 9$, $1\leqslant f\leqslant 9$, $x$, $y$, $e$, $f$ are all positive integers), define: $k=\frac{{F_{(s)}}}{{F_{(t)}}}$. When $F_{(s)}+F_{(t)}=19$, the maximum value of $k$ is ______.

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