
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
A four-digit positive integer $m$ where each digit is not $0$ is called a "twin egg number" if the thousands digit is the same as the units digit, and the hundreds digit is the same as the tens digit. By exchanging the thousands and hundreds digits, and the tens and units digits, a new "twin egg number" $m'$ is obtained. It is defined that $F(m) = \frac{m - m'}{11}$. Find $F(8228) =$ ______; if it is known that the number $m$ is a "twin egg number" and the thousands digit and hundreds digit are different, and $\frac{F(m)}{27}$ is a perfect square, then the smallest value of $m$ that satisfies the conditions is ______.

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