An infinitely long cone can be generated by revolving the following curve about the \(x\)-axis.

\[y = e^{x}, \quad x < 0\]

If the ratio of volume to lateral surface area of the cone is \(\beta \), then what is \( \lfloor 10^4 \times \beta \rfloor \)?

**Notation**: \( \lfloor \cdot \rfloor \) denotes the floor function.

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