
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
During a math activity class, Xiaoming used a rope to form a rectangle with a length to width ratio of $2:1$ and an area of $50 cm^{2}$. <br/>$(1)$ Find the length and width of the rectangle; <br/>$(2)$ He used another rope to form a square with an area equal to that of the above rectangle. Find the difference between the side length of the square and the width of the rectangle.

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