
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 painter uses a straight brush of width w to paint a region along the midline and one diagonal of the square. The painted region covers one third of the square's area. What is the ratio of the length of the square's side to the width of the brush?
A) $2\sqrt{2} - 1$  
B) $2\sqrt{2} + 0.5$  
C) $2\sqrt{2} + 1$  
D) $2\sqrt{2} + 1.5$

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