
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 four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a "naive number." For example, the four-digit number $7311$ is a "naive number" because $7-1=6$ and $3-1=2$. On the other hand, the four-digit number $8421$ is not a "naive number" because $8-1\neq 6$. Find the smallest "naive number" which is ______. 

Let the digit in the thousands place of a "naive number" $M$ be $a$, the digit in the hundreds place be $b$, the digit in the tens place be $c$, and the digit in the units place be $d$. Define $P(M)=3(a+b)+c+d$ and $Q(M)=a-5$. If $\frac{{P(M)}}{{Q(M)}}$ is divisible by $10$, then the maximum value of $M$ that satisfies this condition is ______.

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