\[ \large f(x) = \frac2{e^x - e^{-x}} \left( 1 + \int_1^x f(t) \, dt \right) \]

Suppose a function \(f\) defined on \(x>0\) satisfy the equation above, find the value of \( \displaystyle \left \lfloor 100 \cdot \lim_{a\to\infty} \int_{\frac1a}^a f(t) \, dt \right \rfloor \).

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