Triples of Integers

Find the number of integer triples (a,b,c)(a,b,c) with abca\leq b\leq c and a<c,a<c, such that 288a+b2c2=288b+c2a2=288c+a2b2.288a+b^2c^2=288b+c^2a^2=288c+a^2b^2.

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