We'll take this way too far

Logic Level 3

\[ 2^0 \, \square \, 2^1 \, \square \, 2^2 \, \square \, \ldots \, \square \, 2^{1337} = A \]

There are \(2^{1337}\) ways in which we can fill the squares with \(+, - \). Let \(B\) denote the number of positive integer value of \(A\) such that there exist a solution for the equation above. Find the value of \( \log_2 B \).

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