Suppose that \(0 \leq x \leq 2 \pi \) satisfies \[ \log_{\sin x}{\cos x} + \log_{\cos x}{\tan x} = 1. \] If \(x = \frac{a}{b}\pi\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?

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The two roots of the quadratic equation for \( x \) \[ x^2-(ab-14)x + a^{\log_2 b} - 32 = 0 \] are \( 16\) and \(2 \), where \(0<a<b\). What is the value of \( a + b \)?

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\(x\) and \(y\) are real numbers that satisfy the following two equations: \[ 3^x \cdot 2^y = 3456 \;\;\; \textrm{and} \;\;\; \log_{\sqrt{2}} (y-x) = 4. \] What is the value of \( x + y \)?

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