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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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