
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
Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______.

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