
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]


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In triangle \(ABC\) with side \(BC\) equal to 9, an incircle is drawn, touching side \(BC\) at point \(D\). It is known that \(AD = DC\) and the cosine of angle \(BCA\) equals \(\frac{2}{3}\). Find \(AC\).

Apply the Law of Cosines.

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