
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
In the diagram, $ABC$ is an equilateral triangle where each side is 6 cm. Point $D$ is outside triangle $ABC$, forming another triangle $BCD$ such that $BD = DC = 9$ cm. If $D$ aligns in the same horizontal plane as line segment $BC$ and is directly below point $A$, find the distance $AD$.
A) $9\sqrt{2} \text{ cm}$
B) $6\sqrt{2} + 6 \text{ cm}$
C) $6\sqrt{2} + 3\sqrt{3} \text{ cm}$
D) $3\sqrt{2} + 3\sqrt{3} \text{ cm}$

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