
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: The lengths of the two sides $AB$ and $BC$ of rectangle $ABCD$ are the two real roots of the equation $x^{2}-mx+\frac{m}{2}-\frac{1}{4}=0$ in terms of $x$. <br/>$(1)$ For what value of $m$ is rectangle $ABCD$ a square? Find the length of the side of the square in this case;<br/>$(2)$ If the length of $AB$ is $2$, what is the perimeter of rectangle $ABCD$?

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