
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

In triangle \( ABC \) with angle \( A \) equal to \( 60^\circ \), a bisector \( AD \) is drawn. The radius of the circumcircle around triangle \( ADC \) with center at point \( D \) is \( \frac{2\sqrt{3}}{3} \). Find the length of segment \( BM \), where \( M \) is the intersection point of segments \( AD \) and \( BO \), given that \( AB = 1 \).

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