
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 \(ABC\) be an acute triangle. Let \(DAC\), \(EAB\), and \(FBC\) be isosceles triangles exterior to \(ABC\), with \(DA = DC\), \(EA = EB\), and \(FB = FC\) such that
\[
\angle ADC = 2 \angle BAC, \quad \angle BEA = 2 \angle ABC, \quad \angle CFB = 2 \angle ACB.
\]

Let \(D'\) be the intersection of lines \(DB\) and \(EF\), let \(E'\) be the intersection of lines \(EC\) and \(DF\), and let \(F'\) be the intersection of lines \(FA\) and \(DE\). Find, with proof, the value of the sum
\[
\frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'}.
\]

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