
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
Given an acute triangle \( \triangle ABC \), construct isosceles triangles \( \triangle DAC, \triangle EAB, \triangle FBC \) outside of \( \triangle ABC \) such that \( DA = DC, EA = EB, FB = FC \), and the angles \( \angle ADC = 2 \angle BAC, \angle BEA = 2 \angle ABC, \angle CFB = 2 \angle ACB \). Define \( D' \) as the intersection of line \( DB \) with \( EF \), \( E' \) as the intersection of line \( EC \) with \( DF \), and \( F' \) as the intersection of line \( FA \) with \( DE \). Find the value of \( \frac{DB}{DD'} + \frac{EC}{EE'} + \frac{FA}{FF'} \).

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