Let \( x_1,x_2,x_3,...x_n \) be real numbers such that \( x_1> x_2 > x_3....>x_n \).

Also \( x_1=\tan^{-1} {2} \) and \( \sin (x_{n+1}-x_n)+\dfrac{1}{2^{n+1}} \sin(x_n) \sin(x_{n+1})=0 \) .

Let \( I= \lim _{ n\rightarrow \infty }{ \cot { x_n} } \)

Evaluate \( \large [ {I}] \).

Note : [ ] represents the greatest integer function.

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