
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 natural number $M$ has a non-zero unit digit and can be decomposed into $A\times B\left(A\geqslant B\right)$, where $A$ and $B$ are both two-digit numbers, the tens digits of $A$ and $B$ are the same, and the sum of the unit digits is $6$, then the number $M$ is called a "lucky number", and the process of decomposing the number $M$ into $M=A\times B$ is called a "happy decomposition". For example, because $528=22\times 24$, the tens digits of $22$ and $24$ are the same, and the sum of the unit digits is $6$, so $528$ is a "lucky number".<br/>$(1)$ The smallest "lucky number" is ______;<br/>$(2)$ Decompose a "lucky number" $M$ into "happy decomposition", that is, $M=A\times B$, the sum of $A$ and $B$ is denoted as $P(M)$, and the difference between $A$ and $B$ is denoted as $Q(M)$, if $\frac{P(M)}{Q(M)}$ is divisible by $7$, then the value of $M$ is ______.

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