
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 two shapes. The first shape, Shape $I$, is a square with side $S$, perimeter $P_1$, and area $A_1$. The second shape, Shape $II$, is a circle with radius $r$, circumference $C_2$, and area $A_2$. Determine the relationship between the perimeter of the square and the circumference of the circle.
A) $P_1 : C_2 = 2 : \pi$ always
B) $P_1 : C_2 = 2 : \pi$ only sometimes
C) $P_1 : C_2$ ratio depends on $S$ and $r$
D) $P_1 : C_2 = S : r$ always
E) $P_1 : C_2 = \pi : 2$ always

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