
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 right triangle $BCD$ with right angle at $C$ is constructed externally on the hypotenuse $\overline{BC}$ of an isosceles right triangle $ABC$ where $AB = BC = 2$. If $\triangle ABC$ and $\triangle BCD$ have equal perimeters, determine $\sin(2 \angle BAC)$.
A) $\frac{1}{2}$   
B) $\frac{\sqrt{2}}{2}$   
C) $\frac{3}{4}$  
D) 1  
E) $\frac{\sqrt{3}}{2}$

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