
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
Side \( \overline{AB} \) of \( \triangle ABC \) is the diameter of a semicircle. If \( AB = 3 + \sqrt{3} \), \( BC = 3\sqrt{2} \), and \( AC = 2\sqrt{3} \), then the area of the shaded region can be written as \( \frac{a + (b + c\sqrt{d})\pi}{e} \), where \( a, b, c, d, e \) are integers, \( e \) is positive, \( d \) is square-free, and \( \operatorname{gcd}(a, b, c, e) = 1 \). Find \( 10000a + 1000b + 100c + 10d + e \).

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