
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
On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, and outer sections have $2$ and $5$. The probability of a dart hitting a particular region is still proportional to its area. Calculate the probability that when three darts hit this board, the total score is exactly $12$.
A) $\frac{9}{2048}$ 
B) $\frac{9}{1024}$ 
C) $\frac{18}{2048}$ 
D) $\frac{15}{1024}$

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