
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 the acute triangle \(ABC\), \(AD\), \(DE\), and \(CF\) are altitudes, and \(H\) is the orthocenter. If \(EF\) bisects the area of triangle \(ABC\), find \(DE^2 + EF^2 + FD^2\). (Given the side lengths of the triangle are \(3\), \(2\sqrt{2}\), and \(\sqrt{5}\)).

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