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HMMT 2025 — Algebra & Number Theory Round (Problem 7)

There exists a unique triple (a,b,c)(a,b,c) of positive real numbers that satisfies the equations 2(a2+1)=3(b2+1)=4(c2+1)2(a^2+1)=3(b^2+1)=4(c^2+1) and ab+bc+ca=1ab+bc+ca=1. Compute a+b+ca+b+c.