
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 three-digit positive integer $P$, where the digits in each place are not zero, if the difference between the hundreds digit and the tens digit is equal to the difference between the tens digit and the units digit, then this number $P$ is called a "dream number". For any "dream number", let the sum of its first two digits and last two digits be denoted as $m$; let the sum of the hundreds digit and the units digit forming a two-digit number added to the new two-digit number obtained by swapping these two digits be denoted as $n$; and let the result of dividing the difference between $m$ and $n$ by 9 be denoted as $F(P)$. For example, for $P=246$, since $2-4=4-6$, $246$ is a "dream number". Therefore, $m=24+46=70$, $n=26+62=88$, and $F(246)=\frac{70-88}{9}=-2$. <br/> $(1)$ Please determine if $258$ is a "dream number", and calculate the value of $F(741)$; <br/> $(2)$ If $s$ and $t$ are both "dream numbers", where $s=10x+y+502$, $t=10a+b+200$, $(1\leqslant x\leqslant 9)$, $1\leqslant y\leqslant 7$, $1\leqslant a\leqslant 9$, $1\leqslant b\leqslant 9$, $x$, $y$, $a$, $b$ are all integers), define $k=\frac{F(s)}{F(s)}$, when $2F(s)+F(t)=-1$, find the value of the integer $k$.

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