
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
Let a circle centered at $O$ with radius 2 contain the point $A$. Suppose $\angle AOB = 2\theta$ where $AB$ is tangent at $A$, and $B$ is on the extension of $AO$. If point $D$ is on $\overline{OA}$, but closer to $O$ than $A$, and $\overline{BD}$ bisects $\angle ABO$, find the expression for $OD$ in terms of $s$ and $c$, where $s = \sin 2\theta$ and $c = \cos 2\theta$.

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