Derek's prime-tuple

How many distinct \(4\)-tuples of (positive) primes \((p_1, p_2, p_3, p_4)\) are there such that \(p_1 < p_2 < p_3 < p_4\) and

\[p_1^2 + p_2^2 + p_3^2 + p_4^2 = 2223?\]

This problem is posed by Derek K.

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