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Algebra Level 5

$\huge x^{x^{x+3}} \times x^{2x^{x+2}} = a^{a^{(1-a)/a}}$

For all $x$ and $a$ real numbers that satisfy the equation above, and given that

$\Large x^{-x} \times \left( \frac1a\right)^{-\frac1a}$

can be expressed as $\displaystyle x^b$ , where $b$ is an integer, find the value of $b$ .

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