
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 two-digit number $m$ (neither the tens digit nor the units digit is $0$), if we reverse the digits of this two-digit number $m$ to get a new two-digit number $n$, which is called the "reversed number" of $m$, and place $n$ on the left side of $m$ to form a four-digit number, denoted as ${m'}$, and place $n$ on the right side of $m$ to form another four-digit number, denoted as ${m'}$, define $F(m)=\frac{|{m}'-{m}^{n}|}{99}$. For example, the reversed number of $34$ is $43$, $F(34)=\frac{|4334-3443|}{99}=9$.  
$(1)$ Fill in the blank: $F\left(15\right)=$______;  
$(2)$ If $p=65+a$ ($a$ is an integer, $1\leqslant a\leqslant 4$), $q=30+2b$ ($b$ is an integer, $1\leqslant b\leqslant 4$), the sum of the reversed numbers of $p$ and $q$ is divisible by $9$, find the value of $\frac{F(p)}{F(q)}$.

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