
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

The radius of a circle with center \( O \) is \( 2 \sqrt{5} \). A rectangle \( KLMN \) is inscribed in the sector \( AOB \) with an angle of \( 45^\circ \). The side \( KL \) is situated on the segment \( OA \), vertex \( M \) is on the arc \( AB \), and vertex \( N \) is on the segment \( OB \). Find the sides of the rectangle if one side is twice the length of the other.

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