
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 each digit of a four-digit natural number $M$ is not $0$, and the five times of the digit in the thousandth place is equal to the sum of the digits in the hundredth, tenth, and unit places, then this four-digit number is called a "modest number". For example, for the four-digit number $2163$, since $5\times 2=1+6+3$, $2163$ is a "modest number". Another example is the four-digit number $3147$, since $5\times 3\neq 1+4+7$, $3147$ is not a "modest number". If the four-digit number $\overline{467x}$ is a "modest number", then $x=$______. If a "modest number" $M=\overline{abcd}$ (where $d$ is even), rearrange the tens and units digits of the "modest number" $M$ to the front of the thousand and hundred digits to form a new four-digit number $M'=\overline{cdab}$. Define $G(M)=\frac{M-M'}{99}$. If $G(M)$ is divisible by $11$, and $\overline{abc}$ is divisible by $3$, then the maximum value of $M$ is ______.

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