
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
Inside the triangle \( ABC \), a point \( D \) is chosen such that \(\angle BAD = 60^\circ\) and \(\angle ABC = \angle BCD = 30^\circ\). It is given that \( AB = 15 \) and \( CD = 8 \). Find the length of segment \( AD \). If necessary, round the answer to 0.01 or express it as a common fraction.

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