Logarithmic scales are used when the range of possible values is very wide, such as the intensity of an earthquake or the acidity of a liquid, in order to avoid the use of large numbers.

Find the sum of the roots of the equation

\[ (\log_3x)(\log_4x)(\log_5x) = (\log_3x)(\log_4x) +(\log_4x)(\log_5x) +(\log_3x)(\log_5x).\]

It is given that \(\log_{6}a + \log_{6}b + \log_{6}c = 6\), where \(a\), \(b\) and \(c\) are positive integers that form an increasing geometric sequence and \(b - a\) is the square of an integer.

Find \(a + b + c.\)

The value of \(\displaystyle{x}\) that satisfies the equation \[\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x)\] can be expressed as \(\displaystyle{\log_{a}b}\). What is the value of \(\displaystyle{a+b}\)?

\(\quad\)
**Note**: \(a\) and \(b\) are both fractions and their sum is an **integer**.

\[\large \log_2(\log_{2^x}(\log_{2^y}(2^{1000}))) = 0 \]

If \(x\) and \(y\) are positive integers satisfying the equation above, then find the sum of all possible values of \(x+y\).

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