$a^2 +b^2 +16c^2=9k^2 +1$

Find all prime numbers $$a, b, c$$ and positive integers $$k$$ which satisfy the equation above.

If the solutions can be represented as $(a_1,b_1,c_1,k_1),(a_2,b_2,c_2,k_1),\ldots,(a_n,b_n,c_n,k_n)$

Evaluate $$\displaystyle{{\sum_{i=1}^{n} \left(a_i + b_i + c_i + k_i\right)} }$$.

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