Some simple sequences

The three sequences, $\{ a_n \}_{n=1}^{\infty } , \{ b_n \}_{n=1}^{\infty }$ and $\{ c_n \}_{n=1}^{\infty }$ , satisfy the following:

1. $\displaystyle a_n \leq b_n \leq c_n \quad \forall n \in \mathbb{ N}$

2. $\displaystyle a_n + b_n + c_n = p(n)$ ; which is a polynomial.

3. $\displaystyle a_n b_n c_n$ is a non zero constant $\forall n \in \mathbb{ N}$ .

4. $\displaystyle a_n + \frac{1}{a_{n+1}} + b_n + \frac{1}{b_{n+1}} + c_n + \frac{1}{c_{n+1}} = 0 \quad \forall n \in \mathbb{ N}$ .

5. $\displaystyle {a_n}^3 + {b_n}^3 + {c_n}^3 = 8n^3 + 6n +1$ .

Evaluate: $\displaystyle \lim_{n \to \infty } ( 38n a_n )$