
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 a three-digit number $m$ satisfies that the digits in each place are all different and not zero, then this three-digit number is called a "mutually exclusive number." Remove the unit digit of the "mutually exclusive number" $m$ to get a two-digit number ${m'}$, and let the difference between ${m'}$ and the unit digit of $m$ be denoted as $F(m)$, and the difference between the tens digit and the unit digit of $m$ be denoted as $G(m)$. It is known that a three-digit positive integer $m=20(5x+1)+2y$ (where $x$ and $y$ are integers, and $1\leqslant x\leqslant 9, 1\leqslant y\leqslant 9$) is a "mutually exclusive number," $\frac{{F(m)}}{{G(m)}}$ is an integer and can be divided by $13$. Then the maximum value of the "mutually exclusive number" $m$ that satisfies the conditions is ______.

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