
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 a square \(ABCD\) with side length 2 units such that \(A\) is at the origin and \(B\), \(C\), \(D\) are at \((2,0)\), \((2,2)\), and \((0,2)\) respectively. The particles start simultaneously — one at \(A\) and the other at the midpoint of \(\overline{CD}\), and move at the same speed along the perimeter in the direction \(A\Rightarrow B\Rightarrow C\Rightarrow D\Rightarrow A\). The midpoint of the segment connecting these particles traces out a path that encloses region \(R\). Calculate the ratio of the area of \(R\) to the area of square \(ABCD\).
A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{1}{2}\)
D) \(\frac{2}{3}\)
E) \(\frac{3}{4}\)

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