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Algebra Level 5

Let x,yx, y and zz be real numbers such that

{x2+3xy+3y2=25z2+3yz+3y2=169x(2x+3yz)=3yz. \begin{cases} x^2 + 3xy + 3y^2 & = 25 \\ z^2 + 3yz + 3y^2 & = 169 \\ x( 2x + 3y -z ) & = 3yz. \\ \end{cases}

If xy+yz+xz xy + yz + xz can be expressed as ab a \sqrt{b} , where aa and bb are positive integers, and bb is not divisible by the square of any prime. What is the value of a+ba+b?

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