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system
Determine all values of \(a\) for which the equation \((x - a)^2 - 1 = 2(x + |x|)\) has exactly two distinct roots. Consider two cases:

**Case I:** \(x \geq 0\),
\[
(x - a)^2 - 1 = 4x, \quad x^2 - 2(a + 2)x + a^2 - 1 = 0
\]
The discriminant \(D = 4a + 5\).

The equation has two distinct non-negative roots \(x_{1,2} = a + 2 \pm \sqrt{4a + 5}\), if
\[
4a + 5 > 0 \quad \text{and} \quad a + 2 - \sqrt{4a + 5} \geq 0
\]

The equation has exactly one non-negative root \(x_{1,2} = a + 2 + \sqrt{4a + 5}\), if
\[
D = 4a + 5 = 0, \quad a + 2 \geq 0
\]

**Case II:** \(x < 0\),
\[
(x - a)^2 - 1 = 0
\]
The root \(x_1 = a + 1 < 0\), if \(a < -1\), and the root \(x_2 = a - 1 < 0\), if \(a < 1\).

Thus, the equation has two distinct negative roots \(x_{1,2} = a \pm 1\), if \(a < -1\), and the equation has exactly one negative root \(x = a - 1\), if \(-1 \leq a < 1\).

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