\[ \large z_{n+1} = \dfrac{z_n}{1 + \sqrt{1+z_n^2}} \]

Suppose we define a recurrence relation as described above with initial term of \(z_1 = 1 \). Evaluate the limit below.

\[ \lim_{n\to\infty} \left( 2^{n+1} \times z_n \right) \]

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