\[ \large \frac{dy}{dx} = \frac{ y^{2 - \frac{1}{y}}}{ 1 - \ln \: y} \]

Given that the above differential equation has its solution passing through \((1, 1)\). Find the corresponding value of \(x\) when \( y = e^{\frac{1}{e}} \). Input your answer as \( \lfloor 1000x \rfloor \).

You may use a calculator for the final step of your calculation.

**Clarifications**:

\(e \approx 2.71828\) denotes the Euler's number.

\( \lfloor \cdot \rfloor \) denotes the floor function.

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