# Sum of Products of Subsets

Consider the set $$S=\left\{1, 2, 4, ..., 2^{100}\right\}$$. There are $$2^{101}-1$$ non-empty subsets of this set.

For each subset $$A$$ let $$f(A)$$ be the product of the elements in that subset. For example, if $$A=\{1, 2, 4\}$$ then $$f(A)=8$$.

Find the sum of all $$f(A)$$ as $$A$$ ranges over all the $$2^{101}-1$$ subsets. If the answer when divided by $$126$$ leaves a remainder of $$x$$ then find $$x$$.

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