
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
Two particles move along the perimeter of square ABCD in the direction A⇒B⇒C⇒D⇒A, starting simultaneously but at different speeds. The first particle starts at A and moves at a velocity v, and the second starts at B moving at a velocity 2v. The midpoint of the line segment joining these two particles traces out a region R. What is the ratio of the area of R to the area of square ABCD?

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