
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
Triangles $ABC$ and $BCD$ are equilateral. Additionally, squares $ABEF$ and $CDGH$ are constructed externally on sides $AB$ and $CD$ of triangles $ABC$ and $BCD$, respectively. What is the value of $\overline{EG} \div \overline{BC}$ when expressed in simplest radical form?

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