Dependent progression

Number Theory Level 4

Let $$a$$ and $$b$$ be positive integers. Given that

• $$a,x,y,z,b$$ follows an arithmetic progression if $$x \cdot y \cdot z = 55$$.

• $$a,x,y,z,b$$ follows a harmonic progression if $$x \cdot y \cdot z = \frac{343}{55}$$.

Find the value of $$a^2 + b^2$$.

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