
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
Consider an equilateral triangle \(ABC\), where \(AB = BC = CA = 2011\). Let \(P\) be a point inside \(\triangle ABC\). Draw line segments passing through \(P\) such that \(DE \parallel BC\), \(FG \parallel CA\), and \(HI \parallel AB\). Suppose \(DE : FG : HI = 8 : 7 : 10\). Find \(DE + FG + HI\).

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