
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 the system of equations \((x+ay)^2 = 16a^2\) and \((ax-y)^2 = 4a^2\), where \(a\) is a positive real number. Determine the total area enclosed by these curves.
A) \(\frac{32a^2}{1+a^2+2a}\)
B) \(\frac{16a^2}{1+a^2}\)
C) \(\frac{64a^2}{1+a^2}\)
D) \(\frac{32a^2}{1+a^2}\) - Correct
E) \(\frac{32a}{1+a^2}\)

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