
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
In a triangular region ABC with sides equal to 2a, $\textdollar {\sqrt{2}} \textdollar$ a brush of width w is swept along the height from A to the midpoint of BC, splitting the triangle into two right triangles. If one-third of the area of triangle ABC is painted, what is the ratio of the length of side AB to the brush width, w?
A) 3  
B) 4  
C) 5  
D) 6

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