
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 an ellipse with semi-major axis \( a \) and semi-minor axis \( b \). Its perimeter is approximately given by \( p \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}] \) (using the Ramanujan's approximation for simplicity). The longest line segment (akin to the rectangle's diagonal) connecting two points on the ellipse through the center (essentially the major axis) is \( 2a \). Find the relationship between \( a \) and \( b \) if the perimeter is given to be \( p \) and the "diagonal" is \( 2a \).
**A)** \( b = 2a - \frac{p}{2\pi} \)  
**B)** \( b = 3a - \frac{p}{\pi} \)  
**C)** \( b = a - \frac{p}{3\pi} \)  
**D)** \( b = 4a - \frac{p}{\pi} \)  
**E)** \( b = a + \frac{p}{2\pi} \)

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