
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 $M$, let the digit in the thousands place be $a$, in the hundreds place be $b$, in the tens place be $c$, and in the units place be $d$. The two-digit number formed by the thousands and units digits of $M$ is $A=10a+d$, and the two-digit number formed by the tens and hundreds digits of $M$ is $B=10c+b$. If the difference between $A$ and $B$ is equal to the negative of the sum of the thousands and hundreds digits of $M$, then $M$ is called an "open number." Determine whether $1029$ is an "open number" (fill in "yes" or "no"). If $M$ is an "open number," let $G(M)=\frac{b+13}{c-a-d}$. Find the maximum value of $M$ that satisfies the condition when $G(M)$ is divisible by $7$.

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