
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
If the difference between the thousands digit and the hundreds digit of a four-digit number is $2$, and the difference between the tens digit and the units digit is $4$, then this four-digit number is called a "difference $2$ multiple". If the difference between the thousands digit and the hundreds digit of a four-digit number is $3$, and the difference between the tens digit and the units digit is $6$, then this four-digit number is called a "difference $3$ multiple". Let $p$ and $q$ be numbers that are a "difference $2$ multiple" and a "difference $3$ multiple" respectively. If their units digits are both $3$, and the sum of the digits of $p$ and $q$ are denoted as $G(p)$ and $G(q)$, and $F(p,q)=\frac{p-q}{10}$, if $\frac{F(p,q)}{G(p)-G(q)+3}$ is an integer, then the maximum value of $\frac{G(p)}{G(q)}$ is ____.

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