Calculus Level 5

$\large I = \lim_{k\to\infty} \int_0^1 \dfrac x{2^k} \prod_{n=1}^k \left( e^{x/2^n} + e^{-x/2^n} \right) \, dx$

If $$I$$ can be expressed in the form of $$\dfrac{(e+A)^B}{Ce}$$, where $$A,B$$ and $$C$$ are integers satisfying $$A+B+C<5$$, find the product $$ABC$$.

Clarification: $$e \approx 2.71828$$ denotes the Euler's number.

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