
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 hundreds digit of a four-digit natural number is greater than or equal to the tens digit, and the thousands digit is equal to the sum of the hundreds and tens digits, and the units digit is equal to the difference between the hundreds and tens digits, then we call this four-digit number a "Bachuan Jiaoqing password number". For example, the natural number $4312$, where $3 > 1$, $4 = 3 + 1$, $2 = 3 - 1$, so $4312$ is a "Bachuan Jiaoqing password number". The largest "Bachuan Jiaoqing password number" is ________; if a "Bachuan Jiaoqing password number" $m$ has the thousands digit as $a$, hundreds digit as $b$, tens digit as $c$, and units digit as $d$ ($a$, $b$, $c$, $d$ are integers, and $1\leqslant a,b,c,d\leqslant 9$), the smallest value of $m$ that satisfies the condition that the difference between the number represented by the last three digits of $m$ and 7 times the number represented by the thousands digit of $m$ is divisible by $13$ is ________.

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