
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 \(\triangle ABC\), \(D\) is the midpoint of \(BC\), \(E\) is the foot of the perpendicular from \(A\) to \(BC\), and \(F\) is the foot of the perpendicular from \(D\) to \(AC\). Given that \(BE = 5\), \(EC = 9\), and the area of \(\triangle ABC\) is 84, compute \(|EF|\).

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