
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 $\triangle ABC$, the lengths of the three sides are $a$, $b$, and $c$ with $b < c$, and $AD$ is the internal angle bisector of $\angle A$ with point $D$ on $BC$. 

(1) Find the necessary and sufficient conditions (in terms of $\angle A$, $\angle B$, and $\angle C$) for the existence of points $E$ and $F$ on segments $AB$ and $AC$ respectively (not at the endpoints) such that $BE = CF$ and $\angle BDE = \angle CDF$.

(2) If points $E$ and $F$ exist, express the length of $BE$ in terms of $a$, $b$, and $c$.

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