
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 $D$ be a point on segment $BC$, where $BC$ is parallel to plane $\alpha$. From a fixed point $A$ outside of plane $\alpha$ (with $A$ and $BC$ lying on opposite sides of the plane), draw segments $AB$, $AD$, and $AC$ intersecting plane $\alpha$ at points $E$, $F$, and $G$ respectively. Given that $BC = a$, $AD = b$, and $DF = c$, find the length of segment $EG$.

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