
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 number $M$, if the sum of the digit in the thousands place and the digit in the tens place is equal to the sum of the digit in the hundreds place and the digit in the units place, then the number $M$ is called a "合壁数" (translated as "合壁数" in Chinese). For example, for the number $3465$, since $3+6=4+5$, therefore $3465$ is a "合壁数". For the number $2364$, since $2+6=8$ and $3+4=7$, therefore $2+6\neq 3+4$, hence $2364$ is not a "合壁数". Find the smallest "合壁数" which is ______; if the digits of the "合壁数" $M$ are all distinct and non-zero, swap the digit in the thousands place with the digit in the units place, and swap the digit in the hundreds place with the digit in the tens place to form a new four-digit number denoted as $M'$, if $F(M)=\frac{|M-M'|}{9}$ is a perfect square, then the smallest value of $M$ that satisfies the condition is ______.

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