Solve it Algebraically? I won't dare 2

Algebra Level 2

\[\left| 1-\left| 1-\left| x-1 \right| \right| \right| =\alpha \]

Find the value of \(\alpha\) for which this equation has exactly 4 distinct solutions. If the value of \(\alpha\) is \(A\), then find \(2A\).

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