
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
A square dartboard is enclosed within a larger square with a side length of 4 units. The dartboard itself has been divided using the "hour positions" forming lines at every $45^\circ$ instead of $30^\circ$, creating sectors within the board. If $s$ is the area of one of the four triangular regions such as that between 12 o'clock and 3 o'clock on the dartboard, and $p$ is the area of one of the four outer quadrilaterals between the dartboard and the larger square boundary in the same interval, find the value of $\frac{p}{s}$.
A) 3
B) 6
C) 9
D) 12
E) 15

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