
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
Triangle $I$ is an equilateral triangle with side length $A$, perimeter $P$, area $K$, and circumradius $R$. Triangle $II$ is an isosceles right triangle with equal sides $a$ (legs), perimeter $p$, and area $k$. Calculate and choose the correct statement regarding their properties:
1. $P:p$
2. $K:k$
A) $P:p = \frac{3A}{2a(1 + \frac{\sqrt{2}}{2})}$ 
B) $K:k = \frac{A^2 \sqrt{3}}{2a^2}$
C) $R = \frac{A\sqrt{3}}{3}$ and has no comparison in this problem
D) None of the above are correctly calculated

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