
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 rectangular sheet of paper has side lengths in the ratio $1: \sqrt{2}$. We want to outline a frame of equal width on the sheet such that the area of the frame equals half the area of the paper. How many times should the width of the frame be relative to the shorter side of the sheet?

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