
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
There are two three-digit numbers $m=100a+10b+c$ and $n=100d+10e+f$ $(1\leqslant a,b,c,d,e,f\leqslant 9)$. If $m$ and $n$ satisfy $F(m,n)=\frac{a+4d+4e+4f}{b-c}$ is an integer, then $m$ and $n$ are called the best "partner numbers". Let $p=467+110x$ and $q=200y+z+37$ $(0\leqslant x\leqslant 3, 0 \lt y\leqslant 4, 3 \lt z\leqslant 9)$. If $p$ and $q$ are the best "partner numbers", and the sum of the digits of $q$ is divisible by $12$, then $p=\_\_\_\_\_\_$.

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