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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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