
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
For a circle with center $O$ and radius 2, consider a point $A$ on the circle. Segment $AB$ is tangent to the circle at $A$ and $\angle AOB = 2\theta$. Let $D$ be a point on segment $OA$ such that segment $BD$ bisects $\angle ABO$. Express $OD$ in terms of $t_1 = \sin 2\theta$ and $t_2 = \cos 2\theta$.

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