
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
Let \(AD\), \(BE\), and \(CF\) be segments sharing a common midpoint, forming a \(60^\circ\) angle with each pair, and having lengths \(AD = 7\), \(BE = 10\), and \(CF = 18\). Let \(K\) denote the sum of the areas of the six triangles \(\triangle ABC\), \(\triangle BCD\), \(\triangle CDE\), \(\triangle DEF\), \(\triangle EFA\), and \(\triangle FAB\). Compute \(K\sqrt{3}\).

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