
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
On side \( AB \) of triangle \( ABC \), a point \( D \) is chosen such that \( CD = \sqrt{13} \) and \(\sin \angle ACD : \sin \angle BCD = 4:3\). A line through the midpoint of segment \( CD \) intersects sides \( AC \) and \( BC \) at points \( M \) and \( N \) respectively. Given that \(\angle C = 120^\circ\), the area of triangle \( MCN \) is \( 3\sqrt{3} \), and the distance from point \( M \) to line \( AB \) is twice the distance from point \( N \) to the same line, find the area of triangle \( ABC \).

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