
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
Read the following material and answer the questions. For any three-digit number $n$, if $n$ satisfies that the digits in each place are all different and not zero, then this number is called a "special number". By swapping any two digits of a "special number", we can obtain three different new three-digit numbers. Let $F(n)$ be the quotient of the sum of these three new three-digit numbers divided by $111$. For example, if $n=123$, swapping the hundreds and tens digits gives $213$, swapping the hundreds and ones digits gives $321$, and swapping the tens and ones digits gives $132$. The sum of these three new three-digit numbers is $213+321+132=666$, and since $666\div 111=6$, we have $F(123)=6$.

$(1)$ Calculate: $F(341)=\_\_\_\_\_\_$, $F(625)=\_\_\_\_\_\_$;

$(2)$ If $s$ and $t$ are both "special numbers", where $s=100x+32$, $t=150+y$, $1\leqslant x\leqslant 9$, $1\leqslant y\leqslant 9$, and $x$ and $y$ are positive integers, let $k=F(s)-F(t)$. When $F(s)+F(t)=19$, find the minimum value of $k$.

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