
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
Right triangle $ACD$, with right angle at $C$, is constructed outwards on the hypotenuse $\overline{AC}$ of an isosceles right triangle $ABC$ with leg length $2$. If the two triangles have equal perimeters, what is $\sin(2\angle BAD)$?
**A)** $\frac{7}{25}$  
**B)** $\frac{14}{25}$  
**C)** $\frac{3}{5}$  
**D)** $\frac{21}{25}$  
**E)** $\frac{12}{25}$

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