
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 four-digit positive integer satisfies that the sum of the digit in the thousand's place and the digit in the hundred's place is $k$ times the difference between the digit in the tens place and the digit in the units place ($k$ is an integer), it is called a "$k$ multiple." For example, for the four-digit number $3641$, since $3+6=3\times \left(4-1\right)$, so $3641$ is a "$3$ multiple." If a four-digit number $M$ is a "$4$ multiple," $M-4$ is a "$-3$ multiple," and by swapping the digit in the hundred's place and the tens place of $M$, we get a new four-digit number $N$, which is also a "$4$ multiple." The minimum value of $M$ that satisfies the conditions is ____. Write the minimum value of $M$ as the difference of two positive integers squared, i.e., $M=a^{2}-b^{2}$ (where $a$ and $b$ are positive integers). Among all the square difference decompositions of $M$ with the minimum value, when $a-b$ is minimized, define $F\left(M\right)=\frac{a}{b}$, then the value of $F\left(M\right)$ is ____.

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