Derek's prime-tuple

How many distinct 44-tuples of (positive) primes (p1,p2,p3,p4)(p_1, p_2, p_3, p_4) are there such that p1<p2<p3<p4p_1 < p_2 < p_3 < p_4 and

p12+p22+p32+p42=2223?p_1^2 + p_2^2 + p_3^2 + p_4^2 = 2223?

This problem is posed by Derek K.

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