
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 \(ABC\), side \(BC\) is equal to 4, and angle \(ACB\) is \(\frac{2\pi}{3}\). Circle \(\Gamma\) with radius \(2\sqrt{3}\) is tangent to sides \(BC\) and \(AC\) of triangle \(ABC\) at points \(K\) and \(L\) respectively and intersects side \(AB\) at points \(M\) and \(N\) (\(M\) lies between \(A\) and \(N\)), such that segment \(MK\) is parallel to \(AC\). Find the lengths of segments \(CK\), \(MK\), \(AB\) and the area of triangle \(CMN\).

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