
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 a diagram, $\triangle ABC$ is isosceles with $AB = AC$ and an isosceles triangle $\triangle DEF$ is inscribed within it, where $DE = DF$. Designate $\angle BFD$ as $a$, $\angle ADE$ as $b$, and $\angle FEC$ as $c$. If $\triangle DEF$ has a vertex angle of $100^\circ$, find the relation among $a$, $b$, and $c$.
**A)** $b = \frac{a+c}{2}$ 
**B)** $b = \frac{a-c}{2}$ 
**C)** $a = \frac{b-c}{2}$ 
**D)** $a = \frac{b+c}{2}$ 
**E)** none of these

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