
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
A circle with a radius of 3 is inscribed in triangle \( ABC \) and touches the side \( AC \) at point D. Points \( K \) and \( L \) are taken on the extensions of the sides \( AC \) and \( BC \) beyond point \( C \), respectively, and the angle \( CKL \) is \( 30^\circ \). The lengths of segments \( AK \) and \( BL \) are equal to the semiperimeter of the triangle \( ABC \). A point \( M \) is selected on the circumcircle of triangle \( ABC \) such that \( CM \parallel KL \). Find the cosine of angle \( ACB \) if the length of segment \( DM \) is 6.

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