
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 with area $16$ is inscribed in a square with area $18$, with each vertex of the smaller square touching a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. If the smaller square is rotated by an angle $\theta = 30^\circ$ relative to the larger square, what is the value of $ab$?
A) $4\sqrt{3}$
B) $6\sqrt{3}$
C) $8\sqrt{3}$
D) $10\sqrt{3}$
E) $12\sqrt{3}$

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