Triple the sequence!

Calculus Level 5

\[\large \begin{cases} a_n + b_n + c_n = 2n + 1\\ a_nb_n + b_nc_n + c_na_n = 2n - 1\\ a_nb_nc_n = -1\\ a_n < b_n < c_n \end{cases} \]

Let \(\{a_n\}\), \(\{b_n\}\) and \(\{c_n\}\) be sequences satisfying the system of equations above.

Find the value of \(\displaystyle \lim_{n \to \infty } (n.a_n)\).

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