
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 circle centered at $O$ has a radius of 2 and contains the point $A$ on its circumference. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \phi$. If point $D$ lies on $\overline{OA}$ such that $\overline{BD}$ bisects $\angle ABO$, express $OD$ in terms of the sine and cosine of $\phi$.

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