
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
Consider a square divided into a composition of different geometric shapes: Triangle I, Circle II, and a smaller Square III, each splitting the square differently. Triangle I occupies one half of the square divided diagonally. Circle II is inscribed in one quadrant of the square. Smaller Square III fills a quarter of the main square by side length. Calculate the area ratio of the shaded regions in each configuration to the main square's area.
A) $\frac{1}{2}, \frac{\pi}{4}, \frac{1}{4}$  
B) $\frac{1}{4}, \frac{\pi}{4}, \frac{1}{4}$  
C) $\frac{1}{2}, \frac{\pi}{4}, \frac{1}{2}$  
D) $\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{4}$  
E) $\frac{1}{2}, \frac{1}{2}, \frac{1}{4}$

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