
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
Given triangle $ABC$, which is a right triangle at $C$, we have $\overline{BC} = \frac{1}{2}$, $\overline{AC} = \frac{b}{2}$, and a second triangle $ABD$ is constructed outward such that $D$ is on the same side of $AB$ as $C$ and $\overline{AD} = 1$. Find $\overline{BD}$ if $\angle BAD = \angle BCA$.

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