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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