
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
The interior of a polygon is bounded by the curves $(x - 2ay)^2 = 16a^2$ and $(2ax + y)^2 = 9a^2$, where \( a \) is a positive real number. Determine the area of this region in terms of \( a \), valid for all \( a > 0 \).
**A)** \(\frac{48a^2}{1+4a^2}\)
**B)** \(\frac{24a^2}{1+4a^2}\)
**C)** \(\frac{48a^2}{\sqrt{1 + 16a^4}}\)
**D)** \(\frac{6a^2}{5+20a^2}\)

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