Squared Primes (Olympiad problem)

\[a^2 +b^2 +16c^2=9k^2 +1\]

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

This is from JBMO 2015
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